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Experiment 4: Accelerated Motion of a Freely Falling Body

I. About the Experiment A. Background

alileo was the first to introduce the concept of acceleration; he developed it in describing the motion of falling bodies. As a sixteenth century scientist he was also the first to refute with

observation and experiment Aristotle's falling-bodies hypothesis that objects fell to earth with a rate proportional to their mass. Although Galileo gave no reason other than his observations for why bodies fell with equal accelerations we find that a later researcher by the name of Newton provides the explanation. A falling body is accelerated towards the earth due to the gravitational attraction between the body and the earth. This force of attraction acting on the body is called the weight of the body. Newton deduced that a heavier body was attracted to the earth with more force than a lighter body. The question is, if this is true, why then is Aristotle's hypothesis incorrect? The answer is that the acceleration of a body depends not only on the force, but on the mass as well. The more mass a body has the more tendency the body has to resist a change in its state of motion. This tendency to resist a change in motion was called by Galileo inertia. Newton later incorporated the concept of inertia into the first of his three laws of motion. This law, sometimes called "the law on inertia" states:

Every body continues in its state of rest, or uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it.

As an example of why the Galilean explanation is correct, consider two objects whose weights are 10 and 5 pounds respectively. The 10-pound object is attracted to the earth with twice the force as the 5-pound object. However, the 10-pound object also has twice the inertia, or resistance to a change in its motion, as a 5-pound body. Twice the force acting against twice the inertia produces the same acceleration as half the force acting on half the inertia, therefore, both objects accelerate equally. B. Theory The average velocity of a body is equal to the vector displacement !

r r divided by the time !t

required to travel that distance. In symbols

r v avg =

r r 2 !

r r 1

t2 ! t1="

r r

"t Eqn. 1

The instantaneous velocity v of a body is defined as the limit of this ratio as the increment ∆t is made vanishingly small.

G

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r v ( t) = lim

! t" 0

!r r

!t

# $

% &

=dr r

dt Eqn. 2

When the velocity of a body varies the motion is said to be accelerated. Acceleration is defined as the rate of change of velocity. The average acceleration of a body is

r a avg =

r v 2 !

r v 1

t2 ! t1="r v

"t Eqn. 3

The instantaneous acceleration of a body is defined as the limit of the ratio of change in velocity to change in time as ∆t → 0;

r a ( t) = lim

! t" 0

!r v

!t

# $

% & =

dr v

dt Eqn. 4

For one dimensional motion we can dispense with the vector notation and write v(t) =

ds

dt and a(t) =

dv

dt Eqn. 5

In this experiment we wish to study the motion of a freely falling object starting from rest and falling vertically downward along a straight line. Using the apparatus shown in Figure 2 we will obtain a record of position as a function of time. Since the motion takes place along a straight line, we will dispense with the vector notation in what follows. By using our tabulated values of displacement and time, we can plot a graph of distance versus time, obtaining a curve I similar to that shown in Figure 1. We can see from Equation 2 that the slope of this curve is the instantaneous velocity of the body. The slope is obtained by constructing a tangent to the curve at the point for the instant in question and then determining the slope of the tangent.

Time

Dis

tance

s

t

Curve I

Figure 1

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C.To summarize the main points in the above discussion: 1. The average velocity of a body is obtained by dividing the distance traveled by the time

required to traverse the distance. 2. The instantaneous velocity of an object is the limit approached by the ratio ∆s/∆t as ∆t → 0.

This velocity is also equal to the slope of the tangent to the distance-time curve at the desired point.

3. The acceleration of an object is the time rate of change of its velocity, or a = ∆v/∆t. It is also

the slope of the tangent to the velocity-time curve at the instant considered. 4. For a constant acceleration, the velocity-time curve is a straight line and the average velocity of

the body is also equal to the instantaneous velocity at the mid-point of the time interval used. II. The Experiment A record of the position of the falling bob as a function of time is produced on waxed tape by a series of marks that are made by an electric spark which occurs at the rate of 60 per second.

To High VoltageSparking Apparatus

Six FootStand

Tape

Electromagnet

FreelyFalling

bob

HighVoltage

Wire

Figure 2 By selecting certain marks, measuring their distance from an arbitrary origin and divided by the appropriate time interval we are able to determine the average velocity during that time interval from v =

!s

!t Since the bob is in free-fall and accelerating uniformly, this value of the velocity is also the

instantaneous value of velocity at a time half-way through the time interval.

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The acceleration may now be obtained from the slope of a velocity versus time graph.

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III. Procedure 1. After the instructor has explained the use of the free-fall apparatus, and its dangers, practice

attaching the bob to the electromagnet and dropping it several times, ensuring that the bob does not touch either of the parallel wires during its fall. If the bob brushes one of the wires during its fall, have the instructor check to see if the apparatus is level.

2. Pull a piece of waxed paper into position over the rear wire. Then, with the spark running,

drop the bob and produce a trace. (Note: do not run the spark timer too long without dropping the bob as the waxed paper may catch on fire).

3. Check to see if all the sparks were recorded, if not, or if there is any doubt, check with your

instructor. 4. Place the tape on a table next to a 2 meter stick.

Tape

MeterStick s

1s2

s3

Figure 3 5. Choose the first distinct spark to use as a "zero" position. This means that the time interval

used for the calculations will be 1/60 sec or ∆t = 1/60 sec. Measure the distance in meters to each of the dots from the zero position (s1, s2, s3, etc.). Refer to Figure 3

6. Record your data on the table on the next page. IV. Calculations and Analysis 1. Complete the table by calculating the velocity and the acceleration. Note that velocity is found

by dividing adjacent distances by the time interval and that acceleration is found by dividing adjacent velocities by the time interval. Note that the time interval in each case is 1/60 of a second.

2. Calculate the average acceleration, and record this result on the table. 3. Calculate the % error for your measurement of average acceleration assuming the actual value

is 9.8 m/sec2. %error = 4. Plot a curve showing the relation of distance to time (see Figure 4). At the time 14.5/60

seconds draw a tangent to the distance-time curve.

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5. Plot a curve showing the relation of velocity to time (see Figure 5). Put a "best fit" straight

line on the graph.

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Time Distance Velocity Acceleration (in sec/60) (in meters) (in m/s) (in m/s2) 0 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Average Acceleration -->

6. What value for time do you obtain when you extrapolate your v versus t curve to v = 0 (which

means extend your best fit straight line back to the point where it intercepts the time axis)? Time Value at Horizontal Intercept = What does this time value represent?

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7. a) From the slope of the tangent to the distance versus time curve determine the velocity at

time 14.5/ 60 seconds. Velocity (7a) = b) Find the value of the velocity at this same time from your v versus t graph. Velocity (7b) = c) Next calculate the theoretical value of v at this time (vt = v0 + at) using the value of a (the

acceleration) that you have previously determined. Velocity (7c) =

d) Which of the above methods should be most accurate (a, b, or c)?

Why? V. Questions 1. What would be the appearance of the velocity-time curve if the falling body were so light that

the effect of air friction could not be neglected? Make a sketch and explain. 2. Give an example for each of the following cases: a) Where the acceleration of an object is zero but the velocity is not zero. b) Where the velocity of an object is zero but its acceleration is not zero. 3. If the effects of friction are negligible, which will roll down-hill faster, a Cadillac or a Honda?

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Figure 4

Figure 5

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