Lab: Centripetal Force
For motion along a straight line, a constant net force F
acting on a body of mass m produces a constant acceleration a, related to the
force through Newton's law:
F = ma
Does this relation also hold when the same object is moving
in a circle at a constant speed? As you
have seen in the text, rotational motion with period T on a circle of radius R is a motion
with an acceleration defined by:
a = 4 p2R
T2
Will the magnitude of the force needed to
maintain such a motion be defined by:
F = 4 p2Rm
T2
where m is the same mass as in straight-line motion? The purpose
of this experiment is to find an answer to this question. In the process you
will also learn some general methods of searching for mathematical
relationships that will be useful in later experiments.
The equipment that you will use is shown in Fig. 6-1. It allows
you to measure the force while observing the motion. When the glass tube is swung in a small
circle above your head, the rubber stopper moves around in a horizontal circle
at the end of a string. The
string is threaded through the tube
and fastened to some washers hanging below. The force of gravity on these
washers, acting along the string, provides the centripetal force
needed to keep the stopper moving in a circle. Before taking any measurements,
get a feel for the apparatus. With only one washer on the end of the string to
keep the stopper from getting away, whirl the stopper over your head while
holding onto the string below the tube.
Do you have to increase the pull on the string
when you increase the
speed of the stopper?
What happens if you let go of the string?
The mass of the stopper will remain constant throughout the
experiment. Pull enough string through the tube to allow for a radius of about
1 m. A paper clip attached to the string
just below the tube will serve as a marker to help you keep the radius
constant. (The paper clip must not touch the bottom of the glass
tube.) Once you have set the clip or marked the string, measure the radius
accurately from the center of the stopper to the top of the tube.
Six or more washers hung on the end of the string will
provide an adequate centripetal force. Before any measurements are taken,
practice swinging the stopper so that it revolves in a horizontal plane.
To find the period of revolution of the
stopper, have a partner measure the time and count the number of
revolutions while you swing the stopper around. From the time and number of
revolutions, calculate the period.
Repeat the experiment with larger numbers of washers and
make a table of the various values of
F
(in Newtons) and T
(in seconds).
Here are two ways of comparing your data with the derived
formula. If the formula is valid, then the product FT2 should be constant
for all values of F,
and equal to 4 p2Rm. You can calculate the values of
FT2
from your table and see how close they are to their mean value.
You can also calculate
1/T2 for each value of F and plot F as a function of 1/T2.
(That is, you plot 1/T2 on
the horizontal axis.) If the formula is valid, the graph of
F vs 1/T2
will be a straight line through the origin. The slope of the
line will be
FT2
which will also be equal to 4 p2Rm.
Choose either way to
compare the data with the theory.
If you chose to use the mean value of FT2, within
what percent of 4 p2Rm is the
mean value you obtained?
If you chose to plot F as a function of 1/T2,
within what percent of 4 p2Rm is the slope of the graph
you plotted?
For different values of the radius, the product FT2 should
be proportional to the radius. Check this conclusion by taking a few
measurements with different radii and plot FT2 versus R.
Within what percent of the theoretical value of
the proportionality constant is the proportionality constant you have found?
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