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### EXPERIMENT: MEASURING THE ACCELERATION OF GRAVITY: a_{g}

Aristotle's idea that falling bodies on earth are seeking out their
natural places sounds strange to us today. After all, we know the
answer: It's gravity that makes things fall.
But just what is gravity? Newton tried to give operational
meaning to the idea of gravity by seeking out the laws according to
which it acts. Bodies near the earth fall toward it with a certain
acceleration due to the gravitational "attraction" of the earth. But
how can the earth make a body at a distance fall toward it? How is
the gravitational force transmitted? Has the acceleration due to
gravity always remained the same? These and many other
questions about gravity have yet to be answered satisfactorily.
Performing this experiment, you will become more familiar with the effects of gravity-you find the
acceleration of bodies in free fall yourself and you will learn more
about gravity in later chapters.
### a_{g} by Direct Fall*

In this experiment you measure the acceleration of a falling object.
Since the distance and hence the speed of fall is too small for air
resistance to become important, and since other sources of friction
are very small, the acceleration of the falling weight is very nearly
a_{g}.

### Doing the Experiment

The falling object is an ordinary laboratory hooked weight of at
least 200 g mass. (The drag on the paper strip has too great an
effect on the fall of lighter weights. Even here there is significant drag. See experiment results.) The weight is suspended from
about 3 meters of paper tape. Reinforce the tape by doubling a strip of masking tape over one end and punch a hole in the reinforcement one centimeter from the end.
With careful handling, this can support at least a kilogram weight.
##### *Adapted from R. F. Brinckerhoff and D. S. Taft, Modern Laboratory Experiments in
Physics, by permission of Science Electronics, Inc., Nashua, New Hampshire.

A 110-v timer is set up about 2.5m above the laboratory floor. Students must work in pairs or threes
since one of the group must climb a ladder in order to place the tape in the timer and do the drop.
When the suspended weight is allowed to fall, a 110-v timer will mark equal
time intervals on the tape pulled down after the weight.
The timer has a frequency of 60 vibrations/sec. Such a
small mass affects the timer frequency by much less than 1
vibration/sec. After a few practice runs, you will become expert enough to
mark several feet of tape with a series as the tape is accelerated
past the stationary vibrating timer. This method can be
made to yield a series of dots on the tape without seriously retarding its fall.
### Analyzing Your Tapes

Label with an **A** one of the first dots that is clearly formed near the beginning of
the pattern. Count 5 intervals between dots, and mark the end of the fifth space
with a **B**. Continue marking every sixth dot with a letter throughout the length of
the record, which ought to be at least 2.5 meters long.
At **A**, the tape already had a speed of v_{o}.
From this point to **B**, the tape moved in
a time t, a distance we shall call d_{1} . The
distance d_{1}, is described by the equation
of free fall:

d_{1} = v_{o}t + (a_{g} t^{2})/2

In covering the distance from **A** to **C**, the
tape took a time exactly twice as long,
2t, and fell a distance d_{2} described (on
substituting 2t for t and simplifying) by
the equation:

d_{2}= 2v_{o}t + (4a_{g}t^{2})/2

In the same way the distances **AB**, **AE**,
etc., are described by the equations:

d_{3} = 3v_{o}t + (9a_{g}t^{2})/2

d_{4} = 4v_{o}t + (16a_{g}t^{2})/2

and so on.
All of these distances are measured from
**A**, the arbitrary starting point. To find
the distances fallen in each 6-dot
interval, you must subtract each equation
from the one before it, getting:

**AB** = v_{o}t + (a_{g}t^{2})/2

**BC** = v_{o}t + (3a_{g}t^{2})/2

**CD** = v_{o}t + (5a_{g}t^{2})/2

and **DE** = v_{o}t + (7a_{g}t^{2})/2
From these equations you can see that
the weight falls farther during each time
interval. Moreover, when you subtract
each of these distances, **AB**, **BC**, **CD**, . . .
from the subsequent distance, you find
that the increase in distance fallen is a
constant. That is, each difference **BC** - **AB **
=**CD** - **BC** = **DE** - **CD** = a_{g}t^{2}. This quantity is
the increase in the distance fallen in each
successive 6-dot interval and hence is an
acceleration. Our formula shows that a
body falls with a constant acceleration.
From your measurements of **AB**, **AC**, **AD**,
etc., make a column of **AB**, **BC**, **CD**, **ED**,
etc., and in the next column record the
resulting values of a_{g}t^{2}. The values of a
a_{g}t^{2} should all be equal (within the
accuracy of your measurements). Why?
Make all your measurements as
precisely as you can with the equipment
you are using.
Find the average of all your values of
a_{g}t^{2}, the acceleration in centimeters/(6-
dot interval)^{2}. You want to find the
acceleration in cm/sec^{2}. If you call the
frequency of the tape timer n per second,
then the length of the time interval t is 6/n
seconds. Replacing t of 6-dots by 6/n
seconds gives you the acceleration, a_{g} in
cm/sec^{2}.
The ideal value of a_{g} is close to 9.8
m/sec^{2}, but a small force of friction
impeding a falling object is sufficient to
reduce the observed value by several
percent.
Ql What errors would be introduced by
using a timer whose vibrations are
slower than about 60 vibrations per
second? higher than about 120 vibrations
per second?

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