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Salisbury University Physics Department Physics 121 Laboratory Manual1

Laboratory Exercises of Kinematics, Projectile Motion, Newton’s Laws, and Energy

PHYSICS 121 Laboratory Manual

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INDEX 3

I. ANALYTICAL TOOLS 4

1. RECORDING DATA 4 CALCULATING PRECISION OF THE DATA 5

CONVERSION OF UNITS 6 DATA TABLES 7

2. GRAPHICAL ANALYSIS OF DATA 8 PLOTTING DATA 9 BEST FIT CURVE AND SLOPE 10

3. FURTHER ANALYSIS OF RESULTS 11

II. ORIENTATION 12

1. PHYSICS UNITS FOR THE 121 PHYSICS LAB 12 2. PRECISION AND ACCURACY OF MEASUREMENTS 13

PRECISION 13 ACCURACY 14

3. ERRORS IN THE EXPERIMENT 15 4. SIGNIFICANT FIGURES AND ROUNDING 16

ROUNDING CALCULATED PARAMETERS 17 5. LABORATORY REPORT 18

III. EXPERIMENTS Week of 20

INTRODUCTION LAB EXERCISE (02/06) 20 MOTION GRAPHS (02/13) 22 FALLING OBJECTS (02/20) 24 PROJECTILE MOTION (02/27) 25 NEWTON’S LAWS (03/06) 26 FRICTION (03/13) 27 INCLINED PLANE (03/27) 28 UNIFORM CIRCULAR MOTION (04/03) 29 ENERGY CONSERVATION, PART I (04/10) 30 ENERGY CONSERVATION, PART III (04/17) 31 IMPULSE (04/24) 32

COLLISIONS 33

CENTER OF MASS (05/01) 42 ROTATIONAL MOTION AND MOMENTS OF INERTIA

(05/08) 43

III. ADDITIONAL EXPERIMENTS 44

VECTOR RESOLUTION AND ADDITION 44 FORCES AND WORK ON AN INCLINED PLANE 45 ENERGY CONSERVATION, PART II 46 TORQUE AND CENTER OF MASS 47

III. APPENDICES 48

SAMPLE PRE-LAB QUESTIONS 48 SAMPLE LABORATORY REPORT 49 NUMBERS AND SCIENTIFIC NOTATION 55 SYMBOLS 56 CONVERSION FACTORS 57


I. ANALYTICAL TOOLS 1. Recording Data

The ability to objectively evaluate the reliability of an observation and discipline in recording your observation are of great importance in the investigations you will be making in this course.

Data should always be recorded neatly in columns (or rows) with appropriate headings for each column. The units in which the measurements are made (centimeters, seconds, etc) should appear at the top of each column. Always record raw data and later make necessary computations. If you obtain a datum which you believe is greatly in error, do not erase it. Draw a line though it, but leave it legibly visible. Write a brief note on the reasons why you suspect that particular piece of information to be inaccurate.

Experimental data are always subject to “error.” This experimental uncertainty is imposed by the limitations of the instruments (including our hands and eyes) and the method used for making the measurement. (Ultimately, here is a quantum uncertainty associated with everything we can observe, although we shall not approach that limit in this class!) The experimental “error” can be reduced by increasing the precision of the instruments and using more sophisticated methods but it will always be present. For a particular measurement to be useful it is necessary to evaluate this experimental uncertainty (see section titled “significant figures” for more information).

Suppose several measurements are made of a particular quantity. Because of the limitations imposed by the method of measurement, inability to exactly reproduces experimental conditions, and the instruments used the values of these measurements will differ, i.e., there will be experimental uncertainty. However, if the average value of these measurements is taken a number will be obtained which best represents the measurement. We shall call this the “best value” of the quantity, and designate it M for the mean or average.

Suppose that we measure a large number, N, of data points xi (x sub “i”). Here i =1,2,3,4…N represents the individual data points (that is x1 is the first point, x2 is the second point, and so forth). The average or mean is then given by:

That is “Average equals one over N times the sum of the “x” data points.” The symbol “Σ” is Sigma, the Greek letter “S” and stands for “sum.” This equation is the average you are familiar with – simply add all the data points of the same type and divide by the number of points.

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1. Recording Data (continuation) Calculating Precision of the Data

For a variable that is measured N times, the precision of the data can be determined by calculating the standard deviation or the dispersion as recommended by statistical methods. However, this course is limited in mathematical tools and an approximated way to calculate the dispersion is to calculate the %Difference of the highest and lowest value of the sample, this is

Where average2 is:

The percent difference can be taken as an estimation of the precision of the sample. The lowest the percentage, the higher the precision of the sample, this is, the closer the values to each other. Then, the higher the percentage, the lower precision, this is, the farther apart of the values from each other. If the percent difference is less than 15%, it means that this percentage can be explained by the systematic errors that occur during the experiment. If the percentage is greater than 15%, the dispersion can be explained by extreme lack of care when experiments were performed or the bad use of the math or/and physics. A second question that arises about measurements (experimental values) of a variable is if the average value, or each of the values recorded match the real (theoretical, accurate, well known, etc) value of that variable.

Calculating the Accuracy Then a measurement of the accuracy of the data can be obtained by calculating the %Error:

This calculation can be obtained for each measurement, or just for the average value. Using the same argument as in the case for the calculation of precision, if the percent error is less than 15%, the experimental value is accurate and the percentage can be explained by the systematic errors.

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1. Recording Data (continuation) Conversion of Units

Conversion of units is something that can give a lot of trouble, but which is easy one you learn a few simple rules: 1.The word “per” means “divided by,” so that:dollars per liter means , kilometers per hour means

2.The units should cancel just like numbers in an equation, as in the following example:

3.When you are given a conversion factor, you can use it to make a number equal to one, any number may be multiplied by one, and so to convert:a). 100 centimeters equals 1 meter.

Therefore

So to convert meters to centimeters: 3 m = 3 m × 1 = 3 m × 300 centimeters = 3×102 cm

4.A more complicated example: 1 mile is 1.609 kilometers, therefore:

where we have assumed that 1.609 is a constant and 67 miles is the measured value. Then the correct result should have two significant figures 110 km/h. We noticed that miles had to cancel.

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1. Recording Data (continuation) Data Tables

Before drawing a graph or using the data for calculations using the physics involved, however, we must obtain the data and construct a data table. Below are standard ways in which data may be presented in table form:

Table I: Table II: Table III: Weight force versus Stretch Area vs. Radius Period of a Pendulum vs. Amplitude

The variables listed in each column (or row) in the tables are clearly defined along with the associated units. In a laboratory notebook, the experimental conditions would be clearly described as well. If there is more than one possible independent variable those that are held constant are clearly noted (as in Table III).

In this experiment, we would probably go on to vary Mass or Length while holding the amplitude constant, and we could continue using the same table to record to record our data. The data that are directly observed by our experimental instruments (which may include our eyes and ears!) should always be directly recorded. If any data appear to be erroneous because of a mistake or error, it should be crossed out with a single line and an explanation given for tossing out the data.

Later there will be time to calculate additional quantities from the “raw” data, and additional column or rows, clearly distinguished from the original data, can be used to list the calculated values along with the methods or formulae used in the calculation or references pointing to them (see Table II).

At no time should the reader be confused about what is measured and what is calculated from the measurement. Because the “raw” data is directly recorded, if the calculation turns out to be in error you can always go back and recalculate by a different method.

Mass (kg) Weight (N) (Kg x 9.80m/s2)

Δ length (meters)

0.200 1.96 0.140

0.400 3.92 0.203

0.500 4.90 0.234

0.600 5.88 0.262

0.700 6.86 0.300

0.800 7.84 0.334

0.900 8.82 0.365

Amplitude (degrees)

Mass (kilograms)

Length (m) Period (sec)

5.0 10.0 20.0 30.0 40.0

(constant)0.1500

(constant)0.2050

2.68 2.80 2.78 2.82 2.73

R a d i u s (cm), R 2.5 5.0 10.0

Area (cm2) “measured area”

19.6 78.3 314.0

πR2 (cm2) “calculated area”

19.7 78.5 314.2

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2. Graphical Analysis of Data Scientific data are usually collected by observing some natural activity or planned experiment and then looking for overall

patterns that can be explained by fundamental scientific theories or “laws.” To aid in interpretation, data are usually organized in some way to enhance the possibility that a pattern or relationship that may exist will be evident to the scientist. Patterns, relationships, and generalizations permit deductive reasoning and allow us to make predictions, develop theories, and then design further experiments or observations to test those theories. There is no one best way to organize data to ensure that patterns will emerge, but a visual representation can be one of the best ways to “see” that overall picture. In other words, a graph can be worth 103 tables of numbers.

A graph is simply a visual way of representing our numbers or data. “Graphs” might be one dimensional, as for example a thermometer, a speedometer, or a fuel gauge, the “bar graphs” on the graphic equalizer of your stereo. A medical thermometer gives a visual representation of temperature with the rise and fall of mercury. The change in volume of the fluid is proportional to the change in temperature and so the height of the mercury column changes linearly with temperature. A speedometer has a needle whose angle changes in proportion to the speed of the car and so gives a visual representation of the speed. In both cases, it is often easier to immediately grasp changes in temperature of speed than it would be if we had to first read, then comprehend a series of number, then think about their relative magnitudes.

“Y” versus “X” (Vertical axis versus Horizontal axis)

More commonly, we think of a graph as a two-dimensional picture showing the relationship between two sets of numbers. Usually a horizontal axis (what we often think of as the “x” axis) shows an independent variable, while the vertical axis (often thought of as the “y” axis) shows a dependent variable. The independent variable is a set of data points which we vary experimentally, while the dependent variable is the quantity that we measure as a function of the independent variable: we say “y” is a function of “x” or “y=f(x).” Thus the graph shows “y” versus “x” or “f(x) vs. x.” The variable “y” and “x” might be position as function of time, speed as a function of position, gallons of fuel used as a function of miles driven, etc. In our experiments, we will be able to control one or more variables and will measure other data as a function of those variables. In order to perform a systematic experiment, we will only vary one thing at a time (“x”) while we observe the results (“y,” “z,” etc). A graphical representation of “y(x) versus x” or “z(x) versus x” is often the best way to look for a pattern in the data which will tell us about the science behind our experiment.

x-axis

y-axis

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2. Graphical Analysis of Data (continuation) Plotting Data

Once the data has been carefully recorded in the data tables, graphs are constructed to try to determine patterns which may exist in the data. Here is a suggested procedure: 1.Determine the data range: Most of the data you will see this semester will consist of data very similar to the data in the sample tables shown above (although many of your tables will be much larger!). By a data range, we mean the range from the smallest to the largest values you are going to plot. This will ensure that you can fit all of your data within the limits of your graph. For example, in Table II the range for the area might be from 0 to 315-cm2. The range of the radii would be from 0 to 10-cm.

2.Determine the scales: Make sure that the scales you use are linear – that is the “tick” marks are equally spaced with the same range of values between each. Note also that the size of the scales do not have to be the same for both axes if the ranges are very different. Your scales should be even divisions of the range you decided upon to fit your data.

3.Draw and label the axes: It is very important to clearly label the axes of your graphs (including units!) so that you will know what you have plotted when you review your notes. The graph is labeled “Vertical Variable versus Horizontal Variable.” Thus when you are told to plot Variable A vs. Variable B this means that A goes on the “y axis” while B goes on the “x axis.”

4.Plot the actual data: Mark the positions of your data carefully with dots or crosses, then outline these smaller points with larger symbols such as circles, squares, etc.

Once the data has been carefully plotted, your graph should look like the one shown in the sample of the lab report. As seen from this example, the graph should occupy all the page with the axes labeled and the correct intervals. If you are going to determine the slope, then follow the following steps: (turn the page.)

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2. Graphical Analysis of Data (continuation) Best Fit Curve and Slope

5.Draw the “best fit” curves: One of the most useful aspects of a graph is that it can be used to average random experimental errors made in obtaining the data. Because of these uncertainties, even if theory predicts that it should, your data will rarely fall on a straight line and it looks even worst if you connect the points. It is recommended not connecting the points. However, you can draw a “best fit” straight line representing the data - if the data has a linear relationship. Straight lines are not always the best way to fit data points. A smooth curved line can then be drawn which best fits the data points. For example:

6.Determine the slope: If the data represents a straight line. In the case of the last two graphs, only the left graph represents a straight line. Then, the slope is the tilt or steepness of a straight line defined as change in the vertical units (or “Δy,” where the symbol “Δ” delta is used to mean a change in some value) divided by the change in the horizontal units (“Δx”). We can determine the slope of a straight line by taking any two points on it (not data points, but rather points on the line we have drawn) -redshift-. If the coordinates of any two points on the straight line are (x1,y1) and (x2,y2). If the points are from the left graph then the slope is:

Some times in the exercise, the calculation of a certain parameter is done by using the slope instead of plugging the data into the formula. As in the case of the straight line in the left graph, the meaning of the slope is the acceleration. A way to know what the the slope means is to obtain the units of it. In the case shown here, the units of the rise are m/s, the units of the run are s. Then, the units of the slope are m/s2.

◉ ◉

◉

◉

◉◉

◉

Time(sec)

Velocity(m

/s)

20 3010 40 50 600

42

86

◉◉

◉

◉

◉

◉

◉

Time(sec)

Length(m

)

2 31 4 5 60

0.40.2

0.80.6

10 10

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MAIN MENU3. Further Analysis of Results

Once you have calculated a variable from the physics of the experiment. You need to compare that result with a result obtained with a different experimental method carried during the same experiment or other experiments. Usually, one of the experimental methods is more accurate than the other one. A comparison of both results can be obtained using

Where value1 could be less than value 2. Since this is calculation is an absolute value, the percent difference must be positive always. The average2 is again defined as

This measurement can not tell much about the accuracy of the experiment finding the value of the physical parameter, but this measurement can tell about the precision of both methods with respect to each other. If it is known that one of the methods is more accurate than the other one, the percentage difference reflects the existence of errors during the experiment. Theoretically, an expected value of 0% of the percent difference will indicate that both experiments carried the same result. However, performing an experiment can carry different sources of error. If the value for the percent difference is higher than 15 or 20%, there is a high probability that the experiment was performed poorly or the math and physics were applied wrong by the experimenter, or an external agent was introduced in the experiment without notice. If this value is below or equal to 20 - 15%, the difference can be explained due to the systematic errors. A better way to compare the experimental results is to use the theoretical value. In this case we use the following formula:

If the result of the last calculation is again lower than 20 - 15%, then it is possible to argue that the experimental result is accurate. If this last percent is higher than 15% or 20% means that the experimental value is not accurate.

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II. ORIENTATION 1. Physics units for the 121 laboratory exercises

Units of measurement: All measurements must be in SI units (the metric system) What are the standard units for: Length? Meters (m) Mass? Kilograms (kg) Time? Seconds (s) All your measurements must be converted to the last system of units. An abbreviation for the meter, kilogram and second is MKS. These are the units used on this entire course.

Precision of the instruments: The precision of your instruments is reflected in the amount of significant figures for your measurements. The precision of the instrument is how small the instrument can measure the required parameter. As an example, let us use a ruler. A ruler has 30 cm = 0.3 m = 300 mm. When you measure in centimeters you will have one significant figure after the decimal point, for example 23.4 cm. After converting into meters this reading is 0.234 m, this is three significant figures after the decimal point. Thus, the precision of the ruler is 0.001 m. While in the lab, find the precision for the following instruments:

the meter stick? _____________________ the rulers? 0.001 m the balance? _____________________ the stopwatch?___________________

ALWAYS make your measurement as precisely as possible and BE SURE to record that measurement with the correct number of significant digits. Additionally, always convert units to MKS. Sometimes the instruments are not correctly calibrated. Be sure to have a zero value in your instrument every time you measure a variable.

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2. Precision and Accuracy of Measurements

Precision Imagine that you collected the 15 measurements of the size of a box. These are the measurements:

Table IV. Measurements of the size of a box.

There are two things that are easily noticed from the last table. The first one, is that all the measurements are close to each other. Then, a question that rises after this observation is how close are these measurements? A way to answer this question is to use the standard deviation or dispersion of the data, but these statistical tools are out of the scope of this course. So, instead of calculating the last two values we can calculate the %Difference of the highest and lowest value of the sample, this is

Where average2 is:

The percent difference can be taken as an estimation of the precision of the sample. The lowest the percentage, the higher the precision of the sample, this is, the closer the values to each other. Then, the higher the percentage, the lower precision, this is, the farther apart of the values from each other. In the example, the percent is less than 15%, which means that this percent can be explained by the systematic errors that occur during the experiment. If the percentage is greater than 15%, the dispersion can be explained by extreme lack of care when experiments were performed or the bad use of the math or/and physics. A second question that arises about the measurements of the box is if the average value, or each of the values recorded in the table match the real value of the box.

Size of the

box in meters

1.563 1.565 1.587 1.499 1.601 1.614 1.495 1.567 1.512 1.547 1.497 1.521 1.593 1.555

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2. Precision and Accuracy of Measurements (continuation)

Accuracy Imagine that the real value of the size of the box is 1.567 m and the average value from the table is 1.446 m. The last two values are called the theoretical and experimental value, respectively. Then a measurement of the accuracy of the data can be obtained by calculating the %Error:

This calculation can be obtained for each value in the table, or just for the average value as we did. Using the same argument as in the case for the calculation of precision, if the percent is less than 15%, the experimental value is accurate and the percent can be explained by the systematic errors. Finally, one more observation can be made about Table IV. The values are different. Then, the question that follows after the last reasoning is why the values are different?

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3. Errors in the Experiment

It is well known that errors occur during an experiment. Independently of what these are called, it is more important for the experiment to know what are those errors. Thus, errors that originated at the time to obtain data can be produced by faulty equipment, an uncalibrated system (incorrectly set the zero level of the recording device), and the limitations by the observer. The limitations of the observer are the optical resolution of the eye; the precision of the hands to locate objects and the reaction time to execute a task (such as pressing a button ); and the reaction time of the ear and the range of sound frequencies that the ear can detect, for example. The last type of errors are called systematic errors. However, the focus of understanding this type of errors is not the name of them but the sources of them. There is another type of error that depends on the physical conditions for example, variations of temperature, pressure that may affect the measurements of variables. A way to overcome these errors is to obtain a large amount of measurements of the same variable. The last type of errors are called, random errors. After the data is taken, several mistakes occur in the process of finding the final results. Some of this mistakes are the use of incorrect units. Forgetting to convert to the correct units can cause that all results be wrong. Another example of this kind of errors is if a wrong physics formula is used to calculate a variable. Calculating wrong one of the parameters will affect the other calculations that depend on the first calculation. As a consequence, the final result is wrong. This kind of mistakes are wrongly called calculation errors. However, these are not errors per se since this kind of errors can be controlled by the person doing the experiment. That is why, the last ones are not sources of errors. Finally, another misconception about errors appear when during calculations the parameters are rounded. Rounding variables depend on the significant figures. Rounding variables properly is not a source of error. Because this operation is very well determined by the rounding and significant figures rules, that is why the last one is not a source of error. As a final remark about errors, imagine that a spaceship is sent to Mars. The systematic errors will cause to miss the landing target a few meters from the predicted value, the calculation error will cause the spaceship get lost into space. There is not such a thing as calculation errors.

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4. Significant Figures and Rounding As remainder you have learned in your lecture there are some rules to obtain the correct amount of significant figures when you

are not measuring the variable. As a summary, let us give an example of this procedure. With your calculator you will often get numbers to your calculations that give you many digits plus an exponent if you are using scientific notation. For example:

The significant digits of a number are those that tell you meaningful information. Particularly in experimental science, the number of digits is used to help indicate the uncertainty of the problem. In the example above, both 5.6 and 3.8 have two significant digits – this is basically telling us that we don’t know those numbers to any greater accuracy than that. That is we don’t know if the number 5.6 might really be 5.61 or 5.59 – we are uncertain to that extent. When we do the arithmetic with uncertain numbers than the results must be uncertain to the same degree. Therefore we should report that:

Notice that we have rounded off 1.4737 to 1.5 – the closest two-digit number. If the next digit is above 5 we round up to the higher number, if it is below 5 we round down to the next lower number. If it is exactly 5 (any following digits are also zero) then we round off so that we are left with an even numbered digit – that way we will round up half the time and round down half the time. Thus if we are rounding to three significant digits:

1.4587 → 1.46 3.67298 → 3.67 7.54500 → 7.54 7.53500 → 7.54 1.3009 → 1.30

The Role of ZerosZeros can be either significant digits or they can merely be placeholders. If the zeros are after the numbers they are

counted as significant digits. When we say 1.50000 we mean that we know the number is really 1.50000 and not 1.499 or 1.501, although it could be 1.500001. Zeros that appear before the first non-zero digit are placeholders that tell us what power of ten we have and disappear when we use scientific notation. Thus 0.000150 has three significant digits and in scientific notation would be 1.50x10-4 (Note that we move the decimal point over by four). Numbers larger than 10 can be ambiguous unless scientific notation is used. By 6,000,000 do we mean one significant digit or seven? The lack of a decimal point can indicate that not all seven digits are significant, but it is still confusing. 6.0x106 tell us that in fact we mean only two significant digits.

31

2104737.1108.3

106.5 ×=×

×−

31

2105.1108.3

106.5 ×≅×

×−

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4. Significant Figures and Rounding (continuation) Rounding Calculated Parameters

When doing calculations the significant digits of the answer should agree with the significant digits of the least measured variable you have, that is the one with the least number of significant digits. It does not matter if you know all of the other numbers more accurately; your accuracy is limited by the precision of your instruments.

In order to round correctly, after each operation, the number of significant figures must be obtained, before you start a following calculation. Then, you will use the variable just found with the correct precision (correct number of significant figures) combined with a new measured or calculated variable with the correct amount of significant figures.

As an example, you can follow the calculations section in the sample of the lab report. In the table for Part A, the weight force has three significant figures because the mass is given with three significant figures. As a consequence, the slope of the graph has also three significant figures (400 N/m), because the force and the distance reported with three significant figures.

Finally, the value for the slope, represent the spring constant k. This value is used, to calculate the potential energy of the spring. Because the spring constant and the distance have three significant figures, the value for the potential energy also has three significant figures (0.157 J.).

The rest of the calculations in the sample of the lab report are done in the same way

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5. Laboratory Report: Here is a general format for lab write-ups that you should follow. Some parts may be modified depending on the exercise done, but the components of the laboratory report must be kept.

Name (s)

Full names of each person in the team.

Title of lab and date

Introduction

Describe the goal (s) of the experiment. This is, what you are trying to probe. Mention all the required physics to accomplish the experiment.

Experiment Description

Materials used (including the measuring instruments used)

Description. Describe the adventure of doing the experiment. Make a sketch of the setup and draw the physics parameters involved in the experiment such as forces (FBD), measurements and initial and final positions of the objects involved in the experiment and how they do move. Include the physics formulae used in the calculations. Along with the last description, the following questions may help you to do a good description of the experiment.

What did you measure?

What did you observe?

What did you calculate?

Data. In this section you should record all of your measurements as they come from the instruments. The significant figures of the measurements should be consistent with the precision of the instruments.

Making a data table Refer to the section in the tools section about how to build a table. Include a header with the units and name of variables. Make a consistent table with the correct number of significant figures for all the measurements. Include the trials.

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5. Laboratory Report (continuation):

Results

May be a calculated result or an observed result. Include in this section all your findings using the physics described in the introduction and the experiment description and your calculations. Write your calculations in the following way:

If there are repeated calculations, show one sample for each repeated calculation. If there are not repeated calculations show all of them. (Show the equation, then show the equation with numbers in place including units.) Show the answer with correct units. Your first calculations should be converting measurements to the correct units (MKS). Calculate the precision for repeated measurements. If required calculate the slope of the best fit line. If required calculate here the percentage error or percentage difference. If the experiment is observational. You should record here your observations. Graphs - from time to time you will make graphs. Make a graph using the recommendations given in the tools section. Make your graph using the whole page do not try to save the paper making a tiny graph. Use the sample of the lab report at the end of the manual to help yourself how to do it.

Conclusions

Answer here all the other questions in the procedures for which you do not have to do calculations. Here is the section where you think about the physics. Mention if you accomplished the goal(s) of the experiment.

Talk about the accuracy of your results. This question can be answered based on the precision of your measurements.

Tell how you obtained the amount of significant figures for your results.

Mention why the results are not perfect compared with the theoretical results. Describe the systematic errors.

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III. EXPERIMENTS Introduction Lab Exercise

Pre-Lab Questions: Print the tables from the data page (page 21). During the exercise you will record all measurements on the tables.

A.Measuring Length 1.Using a meter stick, measure and record (on the table) the width of your table top (in millimeters) three times. Convert these measurements to centimeters and meters. 2.With the same meter stick, measure your height (in millimeters). Convert these measurements to centimeters and meters. 3.What was the range in each measurement? (difference from smallest to largest measurement) For table? For height? (Write this answer in Data & Evidence) 4.Which was easier to measure -- the table or your height? Explain. (Write this answer in Conclusions) 5.Was the measurement of the table or your height more precise? why? (Write this answer in Conclusions) 6.Why is important to measure in millimeters first and then convert to meters? (Write this answer in Conclusions)

B. Measuring Mass 7.Make sure your balance is zeroed. If you are not sure how to do this, ask your lab instructor. Measure and record the mass of (5) separate 100g masses. Convert the units to kg. Calculate the average mass of the five masses. Calculate the precision of the data. 8.Now go to the instructor’s table and measure each mass on the electronic balance. Do the same calculations as in step 7. 9.Calculate the percent error of each average in 7 and 8 (What is the theoretical value?) (Write this answer on your table) 10. Based on the last question and the precision of your data, what can you conclude about precision and accuracy of the two measuring instruments? (Write this answer in Conclusions)

C. Variability of repeated measurements 11. Your partner will hold a ruler vertically from the top end. Place your fingers at the bottom end (zero millimeters) so that you can catch the ruler when it is dropped. Make sure that your finger is at the bottom of the ruler at zero millimeters. When you see your partner drop the ruler, catch it between your finger and thumb. Record how far the ruler felt before you caught it. Repeat 10 times until the data table is full. 12. How does the data for the falling ruler illustrate the ideas of precision? (Write this answer in Conclusions) 13. Identify the sources of error in all your experiments. (Write this answer in Conclusions) 14. Based on the precision of your instruments. How many significant figures do you have for parts A, B, and C? (Write this answer in Conclusions)

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TAB

LE

HIG

HT

MA

SSES

RU

LER


Motion Graphs Pre-Lab Questions: For both of the following questions you should use a straight edge to draw lines.

1. Sketch a detailed graph of position vs. time for the following motion: You walk away from the origin at a constant speed for 3.0 seconds to a position at 3.0-m; you stop for 2.0 s; then you return to your starting place by walking at a faster, but still steady, speed of 2.0 m/s. Be sure to label the axes of the graph.

2. Sketch a detailed velocity vs. time graph for the same motion. (Remember to label your axes.) Keep in mind that velocity has a direction and therefore a “sign.” Use away from the origin as the positive direction and towards the origin and the negative direction.

Procedures: Plug in and turn on the Motion Sensor. Turn on the computer and open the LabPro folder on the desktop. Open the Motion Graphs program. Make sure the sensor is recording data and displaying the graph. There should be one large Position vs. Time graph on the screen.

A. Position vs. Time Graphs 1. Make the following graphs on the computer by moving around in front of the motion sensor appropriately. It is not necessary to

have your graph go all the way across the screen. 3 or 4 seconds worth will be OK. On your data page make a column of sketches of the following graphs. To the right of each sketch note what motions you have to make in order to create that graph.

2. In general, how do your walking speed and direction influence the position graphs? Consider that speed may be slow, fast, or zero. Direction may be toward or away from the sensor.

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Motion Graphs (Continuation) B. Velocity vs. Time Graphs

1. Switch the “Position” label to “Velocity” by placing the mouse pointer on the "Position" graph label on the y-axis, holding down the left button, dragging to velocity, and releasing the button.

2. Make the following graphs on the computer by moving around in front of the motion sensor appropriately. NOTE: you will have to move the x-axis upwards on the screen so that you can see the negative part of the graph.

On your data page make a column of sketches of the following graphs. To the right of each sketch note what motions you have to make in order to create that graph.

3. In general, how do your walking speed and direction influence the velocity graphs?

4. Once all the above is finished, close the program. Do not save any changes. Now call your lab instructor to your table for further instructions before you proceed to number 5.

5. Open the “Velocity Match.” Explain in detail, for each time segment, how you will need to move in front of the sensor to match the motion graph. Make sure this is written in your lab report before you go any farther.

6. Now try it. Try it again. Show your best attempt to your instructor. Did you have to do anything differently from what you expected to do? Explain.

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Falling Objects Concepts: • Uniformly Accelerated Motion - Gravity • Graphing – Linear Graphs (Read: I. Tools - 2. Graphical Analysis of Data)

Pre-Lab Questions: 1. What equation can you use to calculate the time it should take the tennis ball to fall from a certain height? 2. Why does a tennis ball fall? 3. Use the equation from 1 to calculate the time for a tennis ball to fall from a height of 4.00 meters. 4. If the ball is dropped from twice as high (i.e., 8.00-m) will it take more than twice as long to fall, less than twice as long to fall, or

exactly twice as long to fall? Check you answer with the appropriate calculation. Answer this question carefully!

Procedures: 1. Set up a data table for all your measurements on a separate page before you begin. (Include a height of 0-m and time of 0-sec

in your data table.) 2. Drop a tennis ball from different heights (ranging from 0-m to 8-m), at least five times from each height; measure and record

the time it takes the ball to hit the ground. How consistent are your data for the various heights? 3. Find the average time for the ball to drop from each height. (The time measurements are repeated at least five times in order to

obtain a more precise answer. You should take the average time as the best value of the time and use it in the rest of your calculations.

4. Make a graph of height vs. average time (remember y vs. x – a graph of height vs. average time means that the height is on the vertical axis and average time is on the horizontal axis.)

5. Calculate (using the equation from pre-lab question 1) the theoretical amount of time for the tennis ball to fall from each height. Graph this data (height vs. theoretical time) on the same graph as in #4.

6. How are the two graphs similar and how are they different? Are their shapes the same? Be specific. What is the best fit (line or curve) for each set of data? (If a graph is linear, it should take a ball dropped twice as far exactly twice as long to hit the ground – a straight line.)

7. Graph height vs. averaged t2/2 for the tennis ball data on a separate graph. 8. Graph height vs. theoretical t2/2 on the same graph as in #7. How do the two graphs compare? Be specific. What is the best fit

(line or curve) for each set of data? What best fit do you expect them to be? Why? 9. Calculate the slope of the best fit line for each set of data in #7 and #8. What are the slope's units? 10.What do the slopes tell you?

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Projectile Motion Concepts: • Projectile Motion • Launch Angle • Velocity Components

Pre–Lab Questions: 1. What equation is used to find the vertical speed v0 of a projectile if the maximum height is known? 2. What vertical height will a projectile rise to if it takes 2.30 seconds from launch until hitting the ground back at the same level?

Show your calculations. 3. What values (variables) must you know to calculate the horizontal distance a projectile will travel?

Procedures: Note: Use the SHORT RANGE setting on the projectile launcher. Do not look directly down the barrel when the launcher is

loaded. The launcher should be firmly clamped to the edge of the table. Record on your paper the launcher’s number. Ask the instructor for help.

A. Vertical Launch 1. Set up the launcher for a vertical launch. Measure the maximum height for the ball to travel up from the launcher’s position at

least 10 times. Use the average height to determine the initial launch velocity (vo). 2. Calculate the initial launch velocity (vo). How good is your result? Can you think of any way to improve the accuracy of the

result?

B. Hit the Target 1. Using vo from part A, calculate the initial velocity components voX and voY for a launch that is 60 degrees above the horizontal. 2. From these, calculate the location where the ball will hit the floor. How accurate do you think this is? 3. Set the launcher for a 60 degrees launch, place the target at the appropriate location on the floor, and see if you hit it. If you do

not, move the target until you do it and record the actual location of the target. 4. Compare your calculated target location with the actual location of the target by calculating the percent difference between the

two values (calculated vs. measured distances).

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Newton’s Laws Concepts • Mass • Force • Weight • Newton’s Second Law

Pre–Lab Questions: 1. Briefly describe Newton’s three laws in your own words. 2. Draw a Free body forces diagram for a friction free cart on a horizontal track with a Tension force applied to it.

Procedures: 1. Attach a string to the cart and run it over the pulley. Use a paper clip on the end of the string to hang masses. 2. Add enough extra paper clips on the hanging end of the string to counter the force of friction holding the cart back. Note: the

cart should not start by itself, but should move with a constant speed when given a slight push. 3. Make a table for all the data you will need to collect below. 4. For these masses (5 g, 10 g, 15 g, 20 g, 25 g) hung on the string measure the amount of time for the cart to move a certain

distance (starting from rest). Make three time measurements for each hanging mass and average the times. How consistent are your data for all measurements?

5. Calculate the acceleration of the cart for each hanging mass using the distance and time data. Calculate the Tension in the string due to the hanging masses for each mass (5 g, 10 g, 15 g, 20 g, 25 g). This tension force is the Applied Force.

6. Make a graph of applied force vs. the resulting acceleration of the cart. Be sure to label each axis with units as well as what is plotted on that axis.

7. On the graph calculate the slope of the best–fit line. What units will the slope have? What does the slope tell you? 8. Repeat the procedures 1-7 above for the cart with extra mass on top of it. 9. What are the main sources of error in this lab? What could you do to decrease the sources of error?

10. Describe how this lab illustrates Newton’s laws of motion. Give examples of how the 3 laws applied to the motion of the cart and of the hanging mass.

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Friction Concepts: • Kinetic Friction • Static Friction

Pre–lab Questions: 1. What is the difference between the coefficient of static friction (µs) and the coefficient of kinetic friction (µk)? 2. What physical parameters do you need to experimentally determine µk? Describe the measurements you need. 3. Besides the physical parameters, what other factors determine the coefficient of friction?

Procedures:

A. Determination of µk 1. Measure the mass of the friction block. Calibrate the force probe. Your lab instructor will outline this procedure during the lab

period. 2. Set the friction block (felt side down) on the level track and attach the force probe to it with the hook. Start the program for

measuring Force. 3. For all Force measurements you will grasp the force probe and move it at a constant speed. The force probe will pull the

friction block that is attached to it by a string. You may keep the force probe in contact with the track, or you may want to hold it just above the track. Use whatever method gives you the best horizontal (constant) line on the graph.

4. Determine the amount of applied force needed to keep the block moving at a constant speed when you pull it with the force probe. How is this applied force related to the force of kinetic friction?

5. Repeat 2 with 100, 200, 300, 400, and 500 g masses added on top of the block. 6. Make a graph of the force of friction versus the normal force of the surface on the block. Find the slope of the best-fit line.

What physical quantity does this value represent? What are the units of this slope? What sources of error are there in determining this value?

7. Why is it easier to determine µk if the object moves with a constant velocity?

B. A Second Determination of µk 1. Repeat the measurements in part A with the narrow side of the block, felt side down, against the track. Do you expect to get

the same or a different value for µk? 2. Complete the graph for this data and calculate the slope as you did in #6. 3. Compare your values for µk from parts A and B (calculate the percent difference). Is this what you expected to find? Explain.

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Inclined Plane Concepts: • Free-Body Diagrams on an Inclined Plane • Tension • Normal Force • Friction • Newton’s Second Law on an Inclined Plane

Pre–lab Questions:

The figure shows the lab set-up, where the arrows show the direction of the acceleration: 1. Draw a free-body diagram for the hanging mass (m2). Write down the sum of forces equation (Newton’s second law) for the

hanging mass.

2. Draw a free-body diagram for the friction block (M1). Write down the two sum of forces equations for the cart.

Procedures: 1. Compare your pre-lab diagrams and equations with your lab group. Reconcile any differences and then check your agreed-

upon answers with the instructor. 2. Set up the inclined track with a pulley at the top. Attach a string to a friction block and hang enough mass on the string so that

the friction block accelerates up the incline. Record the hanging mass. Measure the mass of the friction block. 3. You will need to determine the angle of incline. Make the appropriate measurements and calculate θ. Describe how you did

this in the procedure section of your report. 4. Determine the acceleration of the friction block and hanging mass by releasing the friction block from rest and measuring the

time it takes to travel a certain distance. Do ten time measurements and use the average time to calculate acceleration. 5. Calculate the tension in the string, the force of friction on the block, the normal force on the friction block and the coefficient of

kinetic friction µK. 6. How accurate are your answers compared to your measurements needed to calculate them? What sources of error

contributed the most to the inaccuracy of your answers?

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Uniform Circular Motion

Concepts: • Uniform Circular Motion • Centripetal Acceleration

Pre–lab Question: 1. For an object moving in a circle at a constant speed, in which direction does the centripetal acceleration point? 2. What equation is used to calculate centripetal acceleration?

Procedures:

A. Centripetal Force 1. With the mass hanging freely, adjust the pointer so that it is directly under the mass. Attach the spring to the mass. By rolling

the rotor between the thumb and the fingers, make the mass go in circles so that each time it passes directly over the pointer. Time it for 25 revolutions. (Make three time trials.)

2. Draw the free-body diagram for the mass as it moves around the circular path. In your diagram be sure to indicate what provides the net force on the mass to keep it on the circular path.

3. Compute the centripetal acceleration from your data. From the centripetal acceleration compute the force on the mass that keeps it on the circular path.

B. Weight Force 1. Determine the weight-force necessary to stretch the spring so the mass is directly over the pointer. Draw a free-body diagram

for the mass. 2. The weight-force is equal to the value of the force (the spring force) that provides the centripetal acceleration. Compute the

percent difference between this force and the one calculated from the acceleration in step 2. Which method for determining the force do you think is more accurate? Why?

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Energy Conservation, Part I Concepts: • Mechanical Energy, Conservation of Mechanical Energy

Pre–lab Questions: 1. Qualitatively explain how you would expect the kinetic energy to change as a glider slides down an inclined air track, where

friction is negligible. 2. Qualitatively explain how you would expect the potential energy to change as a glider slides down an inclined air track, where

friction is negligible. 3. If the initial and final mechanical energies are MEi = KEi + PEi and MEf = KEf + PEf, respectively. Assume that the change in

mechanical energy is zero. What relationship do you expect between the change in the kinetic energy and the change in the potential energy?

Procedures:

A. Kinetic Energy 1. Release the glider from rest at the top of the track. What is its speed at the bottom of the track? (Use the kinematics

equations and measure the distance and time. Measure time at least 10 times.) 2. What is the kinetic energy of the glider at the beginning position? At the final position? 3. What is the change in kinetic energy of the glider? Show your calculations.

B. Potential Energy (use the table top as height = 0.000 m) 1. Make the measurements necessary to calculate potential energy and record your data. Draw a diagram of the track showing

where the glider is at the beginning and final positions, and where the height is measured at each position. 2. What is the Initial potential energy of the glider? What is its final potential energy? 3. What is the change in gravitational potential energy of the glider between its initial and final positions? Show your calculations.

C. Conservation of Energy 1. In the same way as in steps A and B, determine the initial and final mechanical energies. Calculate the change in mechanical

energy. What are the sources of error? 2. Calculate the percent difference between the initial and final ME calculated in C1. Is total mechanical energy conserved? 3. Calculate the percent difference between the absolute value of the change in the kinetic energy (A3) and the absolute value of

the change in the potential energy (B3). Do your results agree with your pre–lab predictions? Why or why not?

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Energy Conservation, Part III Concepts: • Non-conservative Forces • Kinetic, Gravitational, and Total Mechanical Energy • Energy Conservation

Pre–lab Question: 1. Qualitatively explain how you would expect the mechanical energy to change as a block slides down an incline, where friction

can no longer be ignored. 2. Write the equation to calculate work.

Procedures: In this part of the lab you will be releasing a friction block from the top of an incline (initial position) and measuring its speed at the

bottom of the incline (final position).

A. Kinetic Energy Release the block from the top of the incline. Use kinematics to determine its speed at the final position. Record the time for at

least 20 trials, use a distance of 1 m that the block travels on the track, and find the final velocity. Calculate the change in kinetic energy of the block. How accurate are your data?

B. Potential Energy Make the necessary measurements to determine the change in gravitational potential energy between the initial and final

positions. Record all of your data. Make a diagram of the incline with the block at both the initial and final positions. Calculate the change in gravitational potential energy between the initial and final positions. How accurate are your data?

C. Conservation of Energy Calculate the total mechanical energy of the block at its initial position and the total mechanical energy of the block at its final

position. Calculate the change in mechanical energy of the block between its initial and final positions.

D. Calculating Work Measure the angle. Using the acceleration of the block and Newton’s second law calculate the nonconservative force. Then,

calculate the work done by the nonconservative force. How is the work done by the nonconservative force compared to the change in mechanical energy? Calculate the percent difference between the work and the change in mechanical energy.

Is mechanical energy conserved? Is total energy conserved? Explain. Does this agree with your pre–lab predictions? Why or why not?

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Impulse

Concepts: • Conservation of Energy • Momentum & Impulse

Pre-lab Questions: 1. A ball is dropped from height H above the table. How fast is it going just before it hits the table (in terms of H and any other

parameters you need)? Hint: Think about energy conservation. 2. A ball bounces from the table to a height D. How fast was it going just after it left the table (in terms of D and any other

parameters you need)? Hint: Think about energy conservation.

Procedures:

A. Observations & Predictions Drop the happy and sad balls from the same height onto the table. Which one do you think experiences a larger impulse? Why?

B. Momentum & Impulse 1. Drop the happy ball from at least 1.000 m above the table. Make the appropriate measurements (30 trials, averaged) to

determine its momentum just before it hits the table and its momentum just after it hits the table and starts to bounce upward. Remember momentum is a vector and has direction.

2. From these, calculate the impulse the table exerts on the happy ball. How accurate do you think this is? 3. Repeat 1 and 2 for the sad ball. 4. Which ball experiences the larger impulse? Explain why.

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Collisions Concepts: • Momentum, Kinetic Energy • Elastic and Inelastic Collisions

Pre–lab Questions: 1. Describe what you think will happen if two gliders of equal mass (on a frictionless air track) collide with each other. Imagine

two scenarios: each glider with different speed; both gliders with the same speed. 2. An object with m1 = 1.0 kg at rest collides with a moving object m2 = 0.5 kg. If the velocity of the first object after the collision

is v’1 = -0.5 m/s and the velocity of the second object after the collision is v’2 = 0.2 m/s. Find the initial velocity of the second object v2.

3. How can you tell whether a collision is elastic or inelastic? What criteria do you use? 4. Print the handout to do the exercise(pp 35-41).

Procedures: Draw arrows in the diagrams for each collision. Each collision is just one single event. There is no repetition. One direction must be positive; the other direction must be negative.

A. Collision between two moving gliders with the same masses 1. Observe and describe a head on collision between the gliders when A has a slow speed and B has a larger speed. 2. How are the final speeds of the two related? Which one is moving faster after the collision? Does this agree with your pre-lab

prediction (#1)? Explain.

B. Collision between gliders with one initially at rest 1. Add an extra mass to each side of glider A. Start with it at rest near the middle of the track and start the other glider at one end

(B). Push glider B towards glider A. Observe and describe the collision. 2. Measure (using the electronic timers) the amount of time for B to pass through the timer gate before the collision and the

amount of time for A and B to pass through the gates after the collision. 3. Determine the speeds of the gliders. 4. Compute the total momentum of the gliders (A + B) before the collision. Compute the total momentum of the gliders (A + B)

after the collision. Compare the total momentum before the collision to the total momentum after the collision by calculating percent difference between them. From your data, was momentum conserved? Explain the reasoning behind your decision.

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Collisions (Continuation) 5. Compute the total kinetic energy of the gliders before the collision. Compute the total kinetic energy of the gliders after the

collision. 6. Compare the total kinetic energy before the collision to the total kinetic energy after the collision by calculating the percent

difference between them. From your data, was kinetic energy conserved? Explain the reasoning behind your decision.

C. Collision between gliders of different mass with different coupling devices 1. Change the central pieces on the gliders from the rubber band and flat edge to the sharp point and wax. Keep the extra mass

on glider A. Leave the glider with the sharp point at rest in the center of the air track and push the other glider toward it. Observe and describe the collision.

2. Measure the necessary times and compute the speeds of the gliders. Determine the total momentum of the gliders before and after the collision and the percent difference between them. Was momentum conserved from your data?

3. Compute total kinetic energy of the gliders before the collision and after the collision. Calculate the percent difference between them. From your data, was kinetic energy conserved?

4. Compare the collisions in B and C. Was either collision elastic? Explain your reasoning using your data

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Center of Mass Concepts: • Center of Mass • Equilibrium

Pre–lab Question: 1. How could you experimentally find the center of mass of a long rod, such as a meter stick? 2. Is the center of mass always exactly in the middle of the rod? Explain.

Procedures: The position where the meter stick and suspended masses balance is the position of the center of mass of the system, xcm.

Theoretically this is given by:

(where xms is the center of mass of the meter stick alone). For each part below, make a diagram showing where the center of mass is; where each mass is; and the amount of each mass.

1. Balance the meter stick alone to find xms, and measure the mass of the meter stick (mms) without the pivot point.

2. Hang m1 (0.1 kg) at 0.15 m, put the pivot at xms, and hang m2 (0.2 kg) at the point necessary to balance the system.

Calculate the theoretical position x2 for the 0.2-kg mass. Compute the percent error in the experimental value. 3. Now, remove the masses and hang only m1 (0.1-kg mass) at 0.05 m, and find the new balance point (xCM).

Calculate the theoretical position for the center of mass. Compute the percent error in the experimental value. 4. Obtain an unknown mass from your instructor. Using the methods of this lab, determine its mass. Compare this to its mass as

measured with a standard balance by finding the percent error. What are the sources of error?

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Rotational Motion and Moments of Inertia Concepts: • Angular Acceleration • Torque • Moment of Inertia

Pre–lab Questions: 1. What is the moment of inertia. Write the equation for the moment of inertia of a system of particles. 2. What are the rotational analogs of Newton’s second law, mass, and acceleration?

Procedures: Set up the apparatus as described by the instructor.

A. Measurements 1. Measure the time (3 trials) for the hanging mass to fall to the floor. Calculate the acceleration of the hanging mass. 2. Determine the radius of the pole by measuring the diameter (rp). From the acceleration and the radius of the pole, determine

the angular acceleration (α) of the rotating apparatus. 3. Use Newton’s second law to calculate the force that the hanging mass exerts on the central pole. (This is the tension in the

string.) Determine the amount of torque this force provides to the rotating apparatus. 4. Calculate the moment of inertia (I) of the rotating apparatus from the angular acceleration and the torque. 5. Remove the masses on top (M), but leave the wing–nuts in place and the hanging mass. Repeat steps 1–4 to determine the

acceleration, angular acceleration, force, torque, and moment of inertia (Io) for the apparatus without the masses. 6. Compute the moment of inertia of the masses alone calculating IM = I – Io. 7. Compare this to the theoretical value IM = 2MD2 by computing the percent difference. Where D is the distance from the pole to

the mass spinning.

B. Observations 1. Observe the rotational motion of the apparatus for a different mass at the same distance from the pole as in part A and record

what you see. Describe the motion and the difference in the angular acceleration and moment of inertia. 2. Observe the rotational motion of the apparatus for the same mass as in part A at a different distance from the pole and record

what you see. Describe the motion and the difference in the angular acceleration and moment of inertia.

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IV. ADDITIONAL EXPERIMENTS Vector Resolution and Addition

Concepts: • Vector Components • Graphical Vector Addition • Component Method of Vector Addition

Pre–Lab Questions: 1. Give the definition of "vector". 2. Will all vectors always have both an x and a y component? Why or why not?

Procedures:

A. Resolution of Vectors into Components 1. Resolve the vector that has a magnitude of 8.50 m at an angle of 55° from the positive x axis into its x and y components

graphically. 2. Determine the components of the vector analytically. 3. Compute the percent differences between the graphical and analytical results for each component.

B. Vector Components 1. Graphically determine the magnitude and direction of the vector with the following components.

x: 22.0 cm y: 44.0 cm

2. Now determine the magnitude and direction analytically. 3. Compute the percent differences between the graphical and analytical results for the magnitude and for the direction.

C. Addition of Vectors 1. Add the following vectors graphically to find the magnitude and direction of the resultant vector.

v1: magnitude=12.3 m/s, θ=35° above the x axis. v2: magnitude=25.6 m/s, θ=105° from the positive x axis.

2. Now add the vectors analytically, using the component method. 3. Compute the percent differences between the graphical and analytical results for the magnitude and for the direction of the

resultant vector. 4. Compare the two methods you used to add vectors. Discuss the strong and weak points of each.

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Forces and Work on an Inclined Plane

Concepts: • Free-Body Diagrams on an Inclined Plane • Tension, Normal Force, Friction • Newton’s Second Law on an Inclined Plane • Work

Pre–lab Questions: The lab set-up will look like:

1 Draw a free-body diagram for the hanging mass (m2). Write down the sum of forces equation (Newton’s second law) for the hanging mass. 2 Draw a free-body diagram for the friction block (M1). Write down the two sum of forces equations for the cart.

Procedures: 1. Compare your pre-lab diagrams and equations with your lab group. Reconcile any differences and then check your agreed-

upon answers with the instructor. 2. Measure the mass of the friction cart. Set up the inclined track with a pulley at the top. Attach a string to a friction cart and

hang enough mass on the string so that the friction cart accelerates up the incline. Record the hanging mass. 3. Determine the acceleration of the cart and hanging mass by releasing the cart from rest and measuring the time it takes to

travel a certain distance. Do ten time measurements and use the average time to calculate acceleration. 4. Calculate the tension in the string. Calculate the force of friction and the normal force on the cart. 5. Calculate the work done on the cart by the force of friction. 6. Calculate the work done on the cart by the tension in the string. 7. Calculate the work done on the cart by the normal force. 8. Calculate the work done on the cart by the force of gravity. 9. The sum of the work done by each force is the net work done on the cart. Determine this value. Compare the net work with

the product of the mass of the cart, the acceleration, and the distance the cart moved (i.e., M1ad). Which do you think is more accurate? Explain.

θm2

M1 aa

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Energy Conservation, Part II Concepts: • Spring Potential Energy • Gravitational Potential Energy • Conservation of Energy

Pre–lab Question: 1. How can you experimentally determine the spring constant of a spring?

Procedures:

A. Determine k 1. Position the plunger of the cart against the end stop of the track. Fasten a string to the cart and place it over a pulley so that

you can hang masses on the string pulling the cart against the end stop. 2. Hang various masses on the string to determine a set of at least five masses necessary to move the cart an adequate range of

measurable distances. Once you determined the masses you wish to use remove all masses from the string and measure the original position of the end of the cart. Now hang each mass on the string and measure the new position of the cart for each case.

3. Make a graph of the applied force versus the compression of the spring (the difference between the compressed position and the original position of the cart) to determine the spring constant k.

B. Spring Potential Energy 1. Measure the compression of the spring when the plunger is completely compressed. 2. Calculate the amount of energy stored in the spring when the plunger is completely compressed. How accurate is your

answer? What are the major sources of error?

C. Conservation of Energy 1. Set up the cart at the bottom of the inclined track with the plunger against the end stop. Release the plunger (tap the release

switch with the mass bar) and measure the maximum position the cart goes up the incline. Repeat your measurements several times and calculate an average final position.

2. From your average final position, determine the change in gravitational potential energy of the cart between this position and its starting position. How accurate is your answer? What are the major sources of error?

3. Calculate the percent difference between the absolute value of the change in the gravitational potential energy (C2) and the spring potential energy (B2). Do your results make sense? Why or why not?

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Torque and Center of Mass Concepts: • Center of Mass • Equilibrium • Torque

Pre–lab Questions: 1. How could you experimentally find the center of mass of a long rod, such as a meter stick? 2. If a meter stick is balanced, what does that tell you about the clockwise and counterclockwise torques?

Procedures: The position where the meter stick and suspended masses balances is the position of the center of mass of the system, xcm.

Theoretically this is given by:

(where xst is the center of mass of the meter stick alone). 1. Balance the meter stick alone to find xst, and measure mst. 2. Hang 100 g at 15 cm, put the pivot at xst, and hang 200 g at the point necessary to balance the system.

a. Calculate the theoretical position for the 200-g mass. Compute the percent error in the experimental value. What are the sources of error?

b. Determine the lever arm for each mass, the clockwise torque, the counterclockwise torque, and the percent difference between the torques.

3. Now, remove the masses and hang only the 100-g mass at 5 cm, and find the new balance point. a. Calculate the theoretical position for the center of mass. Compute the percent error in the experimental value. What are the

sources of error? b. Determine the lever arm for each mass, the clockwise torque, the counterclockwise torque, and the percent difference

between them. This time, the center of mass of the meter stick is not at the pivot point, so the torque provided by gravity on the meter stick must be considered.

4. Obtain an unknown mass from your instructor. Using the methods of this lab, determine its mass. Compare this to its mass as measured with a standard balance.

......

+m+m+m+xm+xm+xm=x

21st

2211ststcm

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V. APPENDICES Sample Pre-laboratory Questions

There are one or two questions that the students have to answer ahead of the exercise as a homework. This is an example how to write the pre-lab questions. Do not type it in the computer, do it by hand.

My Name ____________________

Date______________________

Simple Harmonic Motion Lab:

1. Define the following terms:

amplitude; period; equilibrium position

Amplitude is the distance from an equilibrium position to the maximum displacement of an object such as a swinging pendulum. It can also be found by finding the distance between the maximum and minimum wave displacements and dividing by 2.

Period is the amount of time it takes to complete one cycle of harmonic motion.

Equilibrium position is the position of a mass on a spring or of a pendulum mass where no movement happens when the system "runs down". This is where the weight force and tension force in the spring are equal.

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Sample Laboratory Report Energy Conservation Part II

Lab Partners: Jyn Erso, Cassian Andor, Saw Gerrera, & Chirrut Imwe

Introduction

In this lab we will find the spring constant of the plunger spring in a metal cart, then calculate the spring potential energy and gravitational potential energy to see if energy is conserved. The physics we used was Hook’s Law, potential and mechanical energy.


Equipment: Dynamics cart, metal track, pulley and mass hanger, rulers and meter stick.

In Part A, a plunger (spring) was attached to the cart, which was attached to the end of the track. On the other side of the cart a string was attached in order to hang masses when this string was passed over a pulley with a hanger. The initial position of the cart was measured and then five different masses were hung. The cart moved to five different positions that were also measured as shown for one mass in the figure below.

In order to calculate the spring constant (k) we had to measure a Force and a displacement. The equation with k is : F = -k x.

This force was a weight force from masses hung on a mass hanger with the string run over the pulley. (The weight was calculated by multiplying the hanging mass times acceleration of gravity.) With five sets of points, a graph with Force on the y-axis and displacement on the x-axis was made. Then a “best fit” line was used through the data points to calculate the slope.

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In Part B of the experiment, the cart was pushed against the spring compressing it. The distance that the cart moved in order to fully compress the spring was measured in order to find the potential energy of the spring. To find the potential energy stored in a spring the equation PE = ½ k x2 was used. The next figure shows the compression.

In the last part of the experiment, the plunger was used as a launcher of the cart when it traveled a certain distance on the inclined track. The vertical distance traveled by the cart on the track was measured several times to find the final potential energy of the cart at the highest point. The initial vertical high of the cart on the track was also measured with the spring fully compressed. The initial and final potential energy can be obtained with PE = mc g h. The following figure explains how the cart traveled on the track.

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Results Data:

Part A, the following table shows the spring compressed with the hanging mass

Part B, spring fully compressed

Initial position (m): 0.015 Final position (m): 0.002

Change in position (spring compression in meters): 0.028 Part C, the table shows the cart traveling up the inclined track: Mass of the cart (Kg): 0.4998

Initial height of the cart (m): 0.050 Final height of the cart (m): 0.098

Trial number Final position along track (m)1 0.9032 0.9513 0.8984 0.9635 0.9546 0.9687 0.9808 0.9649 0.95810 0.94411 0.955Average final position 0.955

Trial Hanging mass Weight Force Initial Pos. Final Pos. Δ-Pos.(kg) =mg (N) (meters) (meters) (m)

1 .150 1.47 0.030 0.028 0.0022 .250 2.45 0.030 0.025 0.0053 .350 3.43 0.030 0.023 0.0074 .450 4.41 0.030 0.020 0.0105 .500 4.90 0.030 0.019 0.011

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Results Graph:

0.0160 0.002 0.004 0.006 0.008 0.01 0.012 0.014

5.5

0

1

2

3

4

5

Displacement

Wei

ght F

orce

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Calculations:

For Part A, the values of force and distance were plotted as seen in the previous figure. The slope of the best fit line was calculated as

with units of Newtons divided by meters. These are the units of k, the spring constant.

In Part B with the spring all the way in and locked, the displacement of the spring was 2.8 cm or 0.028 m. Then, the potential energy of the spring is

Then, for Part C the cart’s initial gravitational energy was

The average position the cart traveled up the track was to a final position of 95.5 cm. The height of the front of the cart at this position on the track was 9.8 cm or 0.098 m. The cart’s PE at this point was

The ME at the initial position was the sum of the spring PE and gravitational PEi

The MEf at the final position on the track was the same as the potential gravitational energy at the end, that is, MEf = PEf = mghf = 0.480 J. The cart gained 0.08 J of ME, with the average being ME = 0.4 J. The percent difference between initial and final ME is

m = = 400 4.00 N − 2.00 N

0.009 m − 0.004 mN

m

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Conclusions

For Part A, the sources of error are due to the measurements for the the displacement which are of the order of millimeters. The significant figures come from the measurements in distance which were done using a meter stick. The smallest unit in this instrument is a millimeter. Therefore our significant figures were one after the decimal point for the value of k. As in the previous case, the errors in Part B are due to the measurements in distance when we calculate the compression of the spring. As in Part A, the amount of significant figures for the potential energy of the spring is just one.

For Part C, the percentage difference shows that the cart actually gained mechanical energy. The experiment should shows a loss in mechanical energy, but the amount of errors involved produced an increase in mechanical energy. The dispersion of the data for the cart going up hill on the track can be calculated using the percentage difference between the highest and lowest value which is

This percentage is lower than 15% which shows that our measurements were precise. But some error was introduced when we measured the height at the average of this distance, that is why our measurements are not accurate. The same happened when the initial height of the cart was measured. That is why our result is off from the expected loss of energy. The significant figures for the gravitational potential energy can be obtained by knowing that there are four significant figures after the decimal point in kg for the mass of the cart since our triple balance measures up to a tenth of a gram; and three significant figures after the decimal point in meters for the height of the cart on the track. Therefore, the significant figures for gravitational potential energy are three.

%Diff = × 100 = 8.7% ∣∣∣0.980 m − 0.898 m

0.939 m∣∣∣

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Numbers and Scientific Notation Scientific Notation

In physics we run into both very large and very small numbers. Number can range from small values such as the mass of a proton (0.00000000000000000000000000667 kilograms) to the mass of the Sun (19,870,000,000,000,000,000,000,000,000 kilograms). To save paper and ink, write such big numbers in the form of scientific notation. The form of the numbers is:

1.234×10-56 where the first number is multiplied by ten to the power shown. [Practice this notation on the two numbers given above.] If the exponent (the power to which ten is raised) is negative, this indicates a number that is one divided by the power of ten. The power of ten is often called the order of magnitude of a number. On a calculator you can set the mode to scientific notation and the power of ten is usually entered as “EE” or “10x” for “enter exponent.” Thus:

102 = 10 × 10 = 100 10-2 = 1 ÷ 102 = 1 ÷ 100 = 0.01

5.45×106 = 5,450,000 5.45×10-6 = 5.45 ÷ 1,000,000 = 0.00000545

Generally only a single non-zero digit appears before decimal point, the power of ten can always be chosen to make this happen. To convert to “ordinary” numbers simply take the power to which ten is raised and move the decimal point backwards or forwards that many places. Your calculator probably knows how to use scientific notation but you should know yourself so you can have rough idea of the magnitude of the expected answer and notice if you push the wrong button.

To multiply numbers given in scientific notation, you add the exponents (powers) to which ten is raised and multiply the two numbers. Then the decimal point is moved and the power of ten changed until there is only a single digit in from of the decimal point. For example:

4.32×108 • 7.21×102 = 31.1472×1010 = 3.11472×1011 5.45×10-3 • 3.10×104 = 16.895×101 = 1.6895×102

(Note: Significant digits are not properly handled here: we discuss these later. Also your calculator may do much of this work for you.) Similarly to divide numbers, divide the two numbers and subtract the exponent of the denominator (the bottom number) from that of the numerator (the top number).

Adding and subtracting requires that both numbers have the same power of ten: 3.45×103 + 6.47×102 = (3.45 + 0.647)×103 = 4.097×103

3.45×103 – 4.28×106 = (0.00345 – 4.28)×106 = -4.27655×106 Note that for addition or subtraction, unless both exponents are roughly the same, only the larger number contributes significantly to the sum or difference.

213

2

10518.5105518.01044.41045.2 −− ×=×=×

×

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Mathematical Symbols

= ≠ ≈ ~ ∝ > ≥ >> < ≤ <<

is equal to not equal to approximately equal to on the order of is proportional to greater than greater than or equal to is much greater than is less than less than or eqaul to is much less than

Δx |x| ≅ ≡ lim Δt→0 Σ ∏ n! d/dx ∫

a change in x absolute value of x is almost equal to is defined as limit approaches sum product n*(n-1)*(n-2)*… derivative integral

Symbols

Units in the MKS System

Name Abbrev. Type Equivalent

Ampere Coulomb Farad Henry Joule Kelvin Kilogram Meter Newton Pascal Radians Seconds Tesla Volt Watt Ohm

A C F H J oK kg m N Pa Rad Sec T V W Ω

Electric Current Electric Charge Capacitance Inductance Energy Deg. of temperature Mass Length Force Pressure Angle Time Magnetic Field Electric Potential Power Electric Resistance

A=C/sec A⋅sec C/V J/A2 N⋅m = kg⋅m2/sec2

kg m kg⋅m/sec N/m2 Degrees sec V⋅sec/m2 J/C J/sec = N⋅m/sec V/A

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Conversion Factors Between Different Systems of Units1 m = 100 cm =1000 mm

1 kg = 1000 g

1 m = 39.37 inches = 3.281 feet = 1.094 yards

1 km = 0.6215 miles

1 mile = 5280 feet = 1.609 km

1 inch = 2.540 cm

1 revolution = 2π radians = 360o

1 N = 1 kg m/s2

1 J = 1 N m = 1 kg m2/s2

1 Hz = 1/s

1 kWh = 3.6 MJ

1 eV = 1.602×10-19 J

1 horsepower = 550 ft·lb/sec = 746 W

1 Tesla (T) = 104 Gauss

1 kilogram weighs 2.205 lb

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PHYSICS 121 - salisbury.edu Schedule...ERRORS IN THE EXPERIMENT ... Salisbury University Physics...

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