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Comprehensive Study Guide

For Physics 101 Laboratory Exams

Superposition of Waves Purpose: To obtain the resultant wave formed by the superposition of two traveling

waves. Transverse wave - the medium transmitting the wave vibrates in a direction perpendicular to

the direction in which the wave is moving Longitudinal wave - the medium transmitting the wave vibrates in a direction parallel to the

direction in which the wave is moving Sine wave

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Baseline Trough

Crest

Amplitude

Wavelength

Baseline - the horizontal line drawn through the wave. It represents the undisturbed position of

the medium. Crests - the highest points on the wave above the baseline Troughs - the lowest points on the wave below the baseline Wavelength - (the length of one wave) the distance between two adjacent crests or two adjacent

troughs Amplitude - the distance from the baseline to a crest or trough Frequency - the number of waves passing a stationary point per second Period - the time required for one vibration Period ( T ) and frequency ( f ) are inversely related. That is

. T1for

f1T

Wave speed - the distance that the wave travels divided by the time to travel that distance

fλTλv

where v is the wave speed, is the wavelength, and f is the frequency.

Superposition principle - when more than one wave occupies the same space at the same time,

the displacements add at every point. Constructive interference - when the crest of one wave overlaps the crest of another, their

individual effects add together to produce a wave of increased amplitude. Destructive interference - when the crest of one wave overlaps the trough of another, their

individual effects add together to produce a wave of decreased amplitude

Reinforcement

Figure 1. Constructive Interference.

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Cancellation

Figure 2. Destructive Interference.

Partial Cancellation

Figure 3. Constructive and Destructive Interference.

The Vibrating String Purpose: To study harmonics and overtones in a vibrating string. Standing Waves - are produced by the interference of two wave trains (of the same wavelength,

speed, and amplitude) traveling in opposite directions through the same medium. It is a resonance phenomenon.

Figure 2 below shows the various harmonics and overtones for standing waves on a string. A resonance condition occurs when the tension in the string is adjusted so that displacement nodes occur at each end of the vibrating string as shown below. (Nodes and antinodes are marked with n’s and a’s, respectively, in Figure 2 below.)

Node

Antinodes

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Figure 1. Vibrating String Apparatus.

Electric Oscillator

One Segment

Hanger and Masses

Figure 2. Harmonic Numbers and Overtone Numbers for a Vibrating String.

Fundamental or First Harmonic

Second Overtone orThird Harmonic

nn n n n n n n n a a a a a a

First Overtone orSecond Harmonic

In Figure 1 the tension in the string is equal to the weight of the hanger and its load. This weight is equal to the mass of the hanger and its load times the acceleration due to gravity. In other words the

Tension = mass times acceleration or in symbolic form T = mg, (1) where m is the mass of the hanger and its load in kilograms and g is the acceleration due to gravity (9.8 meters/second2). This tension is adjusted by placing masses on the mass hanger to obtain a particular overtone or harmonic. The wave speed, v, depends on the tension, T, and the linear density of the string, measured in kg/m. Linear density is the mass of the string divided by its length. The wave speed v is given by the equation

v = μ/T (2) where T is measured in Newtons and is measured in kilograms per meter. Since

v = f then f = v/ (3) where is the wavelength and f is the frequency of the wave. This last equation is used to determine the frequency of the electric oscillator. Remember that the frequency of the electric oscillator remains constant in this experiment.

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The Organ Pipe Purpose: To study overtones and harmonics of standing waves in organ pipes. Sound waves in air are longitudinal waves. Longitudinal standing waves can be set up in air columns. Since wind instruments (trumpet, flute, clarinet, pipe organ, etc.) are modified forms of air columns, it is useful to examine the standing wave that can be set up in such columns. These standing waves have nodes and antinodes (marked with n’s and a’s in the figures below). Remember that the wavelength of a wave is twice the distance between adjacent antinodes. Figure 1 shows three cylindrical columns or tubes of air that are open at both ends. Sounds waves originating from a tuning fork travel back and forth within each tube since they are reflected from the ends of the tubes, even though the ends are open. If the frequency of the tuning fork matches one of the natural frequencies of the air columns, the rightward-traveling and leftward-traveling waves combine to form a standing wave and the sound of the tuning fork becomes markedly louder. The patterns in the figure below symbolize the amplitude of the vibrating air molecules at various locations. Wherever the pattern is widest the amplitude of vibration is greatest (a displacement antinode), and wherever the pattern is narrowest, the vibration is zero (a displacement node). Standing waves can also exist in a tube with only one end open, as the patterns in Figure 2 indicate. Here the standing waves have a displacement antinode at the open end and a displacement node at the closed end, where the air molecules are not free to move. A tube open at one end can develop standing waves only at the odd harmonic frequencies. By comparison, a tube open at both ends can develop standing waves at all harmonic frequencies.

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Figure 1. Standing Waves in an Open Organ Pipe.

Figure 2. Standing Waves in a Closed Organ Pipe.

First Overtone orThird Harmonic

Fundamental orFirst Harmonic

Second Overtone orFifth Harmonic

n n n n n

Fundamental orFirst Harmonic

n a a

First Overtone orSecond Harmonic

n n a a a

n n n a a a a

Second Overtone orThird Harmonic

aa a a n a a

Types of Spectra Purpose: To observe the spectrum of light emitted from various types of light sources.

In this experiment only a tiny fraction of the electromagnetic spectrum, that portion which is visible, will be studied using a spectroscope with a diffraction grating. A spectrum formed by a grating is the result of diffraction, the bending of waves around obstacles. Red light is deviated the most and violet light the least. The spectral lines observed in a spectroscope are images of the slit formed by each wavelength. If all wavelengths are present in the light (white light), then the images of the slit overlap and a continuous spectrum (like the rainbow) exists. Continuous spectra arise from incandescent solids, liquids, and even gases when at high pressure. For example, a glowing hot stove and a hot metallic filament of a light bulb have continuous spectra.

A hot gas of low density produces a discrete emission spectrum. When viewed with a spectroscope the images of the spectroscope slit appear as bright lines, each of a separate color. The colors that are emitted by an excited atom are characteristic of that particular kind of atom. They represent a unique set of energy levels (electron orbits) for that kind of atom. A low pressure excited gas either in a terrestrial laboratory or in interstellar space shows a spectrum broken into narrow bright lines, i.e., a discrete emission spectrum. Each chemical substance gives off a characteristic pattern of lines that differ from all others, and this pattern may be used to identify or "fingerprint" that substance.

The theories of atoms, based on quantum mechanics, not classical mechanics, assure us that an atom absorbs the same wavelengths of light it would emit if excited. Thus if a cool gas of a given element is placed between an incandescent object and a spectroscope slit, there will be black lines (regions which receive no light) in the continuous spectrum of the incandescent body. This is an absorption effect, and the pattern of dark lines is called an absorption spectrum.

A fluorescent light contains a low pressure gas mixture of mercury and argon. The mercury gas when excited emits visible and ultraviolet. The visible and ultraviolet light strike the phosphors coating the inside of the fluorescent glass tube. Some of the visible light passes through the phosphors and the tube. The ultraviolet, on the other hand, causes the phosphors to fluoresce producing a mixture of colors that combine to give white. Thus the fluorescent light produces two distinct spectra, one from mercury and one from the phosphors. Types of spectra produced in this experiment: Continuous emission spectrum - produced by the incandescent light bulb. Discrete emission spectrum - produced by the low density gas discharge tubes (hydrogen,

helium, neon, and mercury). Continuous and discrete emission spectra - produced by the fluorescent lamp. Nanometer (nm) - a unit of measurement for wavelengths of light. 1 nm = 10-9 m.

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Slit Wavelength Scale Diffraction Grating Figure 1. Spectroscope Figure 2. Spectroscope Interior

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The Ray Box: Part One Purpose: To learn about reflection and refraction of light as well as color addition and

color subtraction. The law of reflection for light states that when light is incident on a specular (mirror-like) reflecting surface the angle of incidence is equal to the angle of reflection. The diagram below shows a light ray incident on a flat mirror. Notice that the angle of incidence (i) and the angle of reflection (r) are measured from a line that is perpendicular to the surface of the mirror. This perpendicular line is called the normal. The incident ray, reflected ray and the normal line all lie in the same plane. Curved mirror surfaces also obey the law of reflection. The law of refraction for light states that a light ray is bent away from its original direction when it passes from one medium to another if it strikes the interface between the two media at a nonzero angle of incidence. Lenses make use of this property. For example, convex lenses converge light rays, and concave lenses diverge light rays.

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r

Normal

i

MIRROR

Reflected Beam

Incident Beam

Concave f Ray Box Mirror

f Convex Ray Box Mirror

f Ray Box

Convex Lens

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Concave Lens

f Ray Box

The Ray Box: Part Two Purpose: To learn about reflection and refraction of light as well as color addition and

color subtraction. (Read the appropriate sections from the lecture text.) Myopia - nearsightedness or short-sightedness, needs diverging lens for correction Hyperopia - farsightedness or long-sightedness, needs converging lens for correction

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Addition of Colors

Thick Yellow Filter

Ray Box

Yellow White Yellow White

Yellow White

Y W Y

W Y W

Thin Thin Cyan Yellow Filter Filter

White Yellow Green Ray Box

Thin Thin Magenta Yellow

Filter Filter

White Yellow Red Ray Box

Thin Thin Magenta

FilteCyan

r Filter

White Cyan Blue Ray Box

Complementary Colors (Two colors that add to give white)

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Ray Box

Cyan Filter

Red

Screen

Convex Lens

Broad Slit

Filter

Convex Lens

Ray Box

Magenta Filter

Green Filter

Screen

Broad Slit

Yellow Filter

Blue Filter

Screen

Convex Lens

Broad Slit

Ray Box

Addition of Colors

Convex Lens

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Ray Box

Blue Filter

Red

Broad Slit

Screen Magenta

Filter

Convex Lens

Ray Box

Green Filter

Red

Broad Slit

Filter

Screen Yellow

Convex Lens

Ray Box

GreenFilter

Blue

Broad Slit

Filter

Screen Cyan

Telescopes Purpose: To learn about telescopes.

The magnification of a telescope can be found using the focal lengths of its lenses.

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Magnification = =

One of the simplest types of telescopes (Figure 1) involves two convex lenses and gives

an inverted image. The lens nearest the observer's eye is called the eyepiece and the other is the objective. The objective lens produces a small, real image of an object being viewed (Figure 2 in the lab manual). For distant objects this image is just outside the focal point of the objective lens. The eyepiece lens is used as a simple magnifying glass to observe the image formed by the objective lens (Figure 3 in the lab manual). (The eyepiece must be moved so that the image produced by the objective falls within the focal point of the eyepiece.) This image formed by the eyepiece is a virtual image of the real image produced by the objective. The observer will see an inverted image with this type of telescope. This type of telescope is commonly known as an astronomical telescope. A third lens (Figure 4 in the lab manual) can be introduced between the objective and the eyepiece in such a way as to produce an upright image for the observer. This lens (inverter lens) effectively transforms the astronomical telescope into a terrestrial telescope.

Another type of telescope is the opera glass (Figure 5). It involves a convex lens (Figure 6 in the lab manual) as the objective and a concave lens as the eyepiece. The resultant image from this combination presents an erect, magnified, and virtual image to the eye of the observer (Figure 7 in the lab manual). The field of view is quite limited, which is okay for operas but is not very useful for watching football games for example.

Important Terms Focal point - the location of the focus of parallel light rays reflecting from a concave mirror

(converging mirror) or passing through a convex lens (converging lens). It is also the location from which light appears to originate for a convex mirror (diverging mirror) that is reflecting parallel light rays or a concave lens (diverging lens) that is diverging parallel light rays. (See figures in Ray Box: Part One.)

Real image - image that can be viewed by the reflection from a screen placed at the image’s location. The light rays forming the image actually pass through the physical location of the image.

Virtual images - the image cannot be formed on a screen. The light rays forming the image only appear to pass through the physical location of the image.

Concave lens - a lens that diverges light. Sometimes called a diverging or negative lens. Convex lens - a lens that converges light. Sometimes called a converging or positive lens.

Focal Length of the Objective Lens fo

fe Focal Length of the Eyepiece Lens

Astronomical Telescope

Figure 1. Astronomical Telescope.

Distant Object (An arrow)

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Galilean Telescope (Opera Glass)

30

fo fo fe+ +

fe

Eyepiece (C s) onvex Len

Objective

Eye

(Convex Lens)

fo fo

Objective (Convex Lens)

Figure 5. Galilean Telescope (Opera Glass).

Distant Object (An arrow)

Eyepiece (Concave Lens)

+fefe

+

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Graphing Purpose: To teach the student how to analyze data graphically.

One of the major problems of physics is to take sometimes seemingly unrelated sets of numbers (data) and make sense out of them. Often the easiest way to do this is by graphing the data. An example of graphing techniques follows.

When weight is added to a spring hanging from the ceiling, the spring stretches. How

much it stretches depends on how much weight is added. The following data was obtained by adding several different amounts of weight to the spring and measuring the corresponding stretch.

Stretch Weight (meters) (Newtons) 0.1240 6.0 0.1475 14.0 0.1775 22.0 0.1950 30.0 0.2195 38.0 0.2300 40.0 0.2525 47.0 0.2675 54.0 0.2875 58.0

There are two variables or parameters that can change during the experiment, weight and

stretch. The experimenter controls the amount of weight to be added. The weight is therefore called the independent variable. The amount that the spring stretches depends on how much weight is added. Hence the stretch is called the dependent variable. The dependent variable is the quantity that depends on the independent variable.

A graph of this experiment is shown in Figure 1. The independent variable is always

plotted on the horizontal axis (called the abscissa). The dependent variable is plotted on the vertical axis (called the ordinate).

If there is a general trend to the data, then a best-fit curve describing this trend can be

drawn. In the example above the data points approximately fall along a straight line. This implies a linear relationship between the stretch and the weight. Note that drawing a best-fit curve does not mean connecting the dots, but rather drawing a smooth curve. This line or curve does not need to "hit" all the data points, as seen in Figure 1.

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Figure 1. Example Graph. Figure 1. Example Graph.

0 10 20 30 40 50 60 Weight (Newtons)

Stretch Versus Weight St

retc

h (m

eter

s)

0.30 0.26 0.22 0.18 0.14 0.10

Newton/meters00294.0Newtons0.17meters05.0

runriseslope

rise = (0.21 - 0.16) meters = 0.05 meters

run = (34.0 - 17.0) Newtons = 17.0 Newtons

y-intercept = 0.11 meters

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Finding the Equation that Describes the Data

If data points follow a linear relationship (straight line), the equation describing this line is of the form y = mx + b, where y represents the dependent variable (in this case, stretch), and x represents the independent variable (weight). The value of the dependent variable when x = 0 is given by b and is known as the y-intercept. The y-intercept is found graphically by finding the intersection of the y-axis (x = 0) and the smooth curve through the data points. In this case b = 0.1060 m. The quantity m is the slope of the best-fit line. It is found by taking any two points on the straight line drawn through the data points and subtracting their respective x and y values using Equation 1.

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12

xxyym

. (1)

If you use two of your circled data points to find the slope, then these points must lie

exactly on the best-fit line. It is easy to determine the slope from the graph. Take any two points on the best-fit data line and draw a triangle like the one shown for this experiment in Figure 1. Measure the lengths of the horizontal and vertical components of the triangle using the units of their respective axes. Then divide the vertical length (amount of rise) by the horizontal length (amount of run). This gives the familiar rise/run formula and is exactly the same as Equation 1 described above. The math for finding the slope in this experiment is given in Figure 1.

At this point everything needed to write the equation describing the data has been found.

Recall that this equation is of the form, y = mx + b. (2)

For Figure 1, y (the dependent variable) is the Stretch (measured in meters), m (the slope which is the rise/run) is 0.00294 meters/Newton, x (the independent variable) is the Weight (measured in Newtons), and b (the y-intercept) is 0.11 meters. Therefore Equation 2 become Stretch = (0.00294 meters/Newton) Weight + (0.11 meters). (3) (y = m x + b) (Note that the “” in Equation 3 represents multiplication.)

In this example one over the slope of this curve is called the spring constant of the

spring. This method of determining the spring constant of a spring is better than alternate methods such as calculating the spring constants of individual measurements and taking an average or taking an average of weights and dividing by an average of the stretches. Graphing is a powerful analytical tool.

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The Simple Pendulum Purpose: To use a simple pendulum to calculate the acceleration due to gravity. Recall from lecture that a pendulum will execute simple harmonic motion for small amplitude vibrations. The period is the time that it takes for the pendulum bob to make one oscillation, while the frequency is the number of oscillations the pendulum makes per second. Period (T) and frequency (f) are reciprocals of one another; that is,

T = 1f or f = T

1 .

The period of a pendulum is independent of the

mass but depends on its length and its amplitude (angle of swing). If the displacement angle is small (less than 10), then the period of the pendulum depends primarily on the length (l ) and the acceleration due to gravity (g) or

T = 2g .

quaring both sides of the equation yields S

.g

4T2

2

This can be graphed with as the independent variable and T2 as the dependent

variabhe local

le. The y-intercept should be zero and the slope should be 42/g. Therefore, the

acceleration due to gravity can be determined from the slope. (For comparisons assume tvalue of g is 9.81 m/s2 or 981 cm/s2.) [Note: Pendulums are used in a variety of applications from timing devices like clocks and metronomes to oil prospecting devices.]

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Addition of Vectors Purpose: To learn about the addition of vectors.

Recall that a vector is a physical quantity that has a magnitude and an associated direction. In other words a vector has both a magnitude (length) and a direction (angle measured from a defined direction). One way of adding vectors is to represent them with arrows that have length and direction. Consider the three vectors below that are to be added.

45 @ N 3F1

N

Notice that the vectors are drawn on a set of axes that look like a compass - north is up, south is down, east is right, and west is left. The first force vector can be specified as having a magnitude of 3 Newtons and a direction of 45° north of east. The second force vector has a magnitude of 1 Newton and a direction of 60° north of east. The third has a magnitude of 2 Newtons and a direction of 210° north of east (or 30° south of west). The resultant (vector sum) of these three force vectors is written as

321 FFFR

.

Notice that the arrow above each letter is a reminder that the quantity has both magnitude and direction. The resultant of these three vectors can be measured experimentally by using the "head-to-tail" method. (Refer to the figure on the next page.) Draw all the vectors sticking the tail (blunt end) of one vector to the head (pointed end) of another. Do this in several steps. First, add 1F

and 2F

. To do this, draw 1F

and then stick 2F

on the head of 1F

. Always keep each

vector in its proper orientation in space as shown in the figure. The resultant 'R

of 1F

and 2F

is the vector drawn from the tail of 1F

to the head of 2F

. The length of 'R

and its orientation in

space represent the vector sum of 1F

and 2F

. To add to F3F

1

and 2F

, place 3F

at the head of 2F

and connect the tail of 1F

with the

head of . This is the resultant of F3F

1

+ F2

+ 3F

and is indicated by R

. The resultant is

measured to be 2.2 Newtons. The direction of the resultant is measured with the protractor to be 66° north of east.

S

E W

60 @ N 1F2

210 @ N 2F3

2F

1F

'R

2F

1F

R

3F

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The resultant can be checked experimentally on an apparatus called a force table (shown to the right). On the force table four mass hangers are attached to cords passing over pulleys mounted on the edge of a horizontal circular metal table. The rim of this table has a scale marked off in degrees. The four cords are hung from a small ring slipped loosely over a post at the center of the table. To do vector addition on the force table for the three vectors mentioned in the above example, designate a mass hanger to be

1F

2F

3F

, a different mass hanger to be , and a third mass hanger to be . Move each cord to the angle for the vector it represents. Add the appropriate masses to each hanger so that the weight of the hanger and its load equal the magnitude of the force that is being represented. Below is shown the force table (top view) with the cords in the proper positions.

E

The line labeled is a quantity called the equilibrant. Its magnitude is 2.2 Newtons, and its direction is 180° from the resultant. In other words, the equilibrant is the value of the vector that just balances the three vectors. The resultant is the sum of the three vectors (shown as dotted). If the equilibrant balances the resultant, then the resultant force on the ring is zero. So when the three vectors to be added are placed properly with respect to the equilibrant and the small pin at the center of the table is removed, then none of the weights move.

R 1F

3F

2F

E

Force Table

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Examples of the Addition of Vectors

Yellow is the addition of the red and blue. Yellow is the addition of the red, blue, and cyan.

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Linear Momentum Purpose: To demonstrate and verify the vector nature and conservation of linear

momentum.

The momentum of a particle is defined as the product of the mass of the particle times its velocity (a vector quantity which obeys the rules of vector addition). That is

where p is the momentum of the particle, m is the mass of the particle and v is its velocity. As long as there are no external forces acting on a system of particles, collisions between the particles will exhibit conservation of linear momentum. This means that the vector sum of the momenta before collision is equal to the vector sum of the momenta of the particles after collision.

This is easily demonstrated in a simple experiment where two objects collide in a horizontal plane and then undergo projectile motion after the collision. Since the horizontal component of velocity remains constant for a projectile in free fall, the horizontal part of the projectile motion can be used to represent the horizontal component of the momentum after collision.

Figure 1. Linear Momentum Apparatus.

v mp

Ball A

Ball B

Plumb Bob

Paper

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Centripetal Force

Purpose: To learn about uniform circular motion and centripetal force. Below is a diagram of the centripetal force apparatus:

MassesSlotted

SpringBob

Pulley String

Index

Radius

Figure 1. Centripetal Force Apparatus.

The spiral spring exerts the centripetal (center seeking) force on a bob that travels in a horizontal circle. The force is found from knowing the mass (M) of the bob, its angular velocity () and its radius of rotation (R). The force can be experimentally checked by direct measurement of the tension in the spring. Important Terms Radian - is an angle unit based on an angle measurement that is the ratio of the arc length of a

circle divided by the radius of the circle. A full circle has an arc length of 2r. Divide that by r to get 2which would be 2 radians. That means 2 radians = 3600.

Centripetal force - is a center-seeking force and is the kind of force necessary to make a object deviate from traveling a straight line.

Angular velocity is 2 radians divided by the time for one revolution.

Comprehensive Study GuideFor Physics 101 Laboratory ExamsSuperposition of WavesThe Vibrating String The Organ PipeTypes of SpectraThe Ray Box: Part OneThe Ray Box: Part TwoTelescopesGraphingThe Simple PendulumAddition of VectorsLinear Momentum Centripetal Force