5 Newton's Second Law II : Uniform Circular Motion

5 Newton’s Second Law II : Uniform Circular Motion
Introduction
A special application of Newton’s Laws involves objects constrained to move along circular
paths. In this case, we know the acceleration of the system must be the centripetal
acceleration, given by
a = v2
r
and the radial component of Newton’s second law takes the form
(1)
m v2
r = Fcent (2)
where F cent is the net force acting toward the center of the circular path. In this lab, you will
have a chance to apply this model to a mass moving with constant speed and constrained
to a circular orbit by a spring.
• First Caution
There is no such thing as “the centrifugal force.”
Experience tells us that when we travel along a circular path, we feel a “force” pushing
us radially outward. For example, when you turn left in a car, you feel a pull to the
right. You might be tempted to call this the “centrifugal force” and include it in the
sum on the right side of Eq. 2. Resist this temptation! This pull you feel is not a
force. It is nothing more than Newton’s first law at work – the tendency of your body
to continue moving in a straight line.
There are centrifugal forces – forces which act away from the center of a circular
path. For example, if a satellite in orbit fires a rocket toward the center of its orbit, a
force is exerted on the satellite directed radially outward. This is an example of a real
force which must be included in the sum on the right side of Eq. 2.
• Second Caution
There is no such thing as “the centripetal force.”
There is no fundamental force of nature called “the centripetal force.” The term
centripetal comes fromalatinterm meaning “center-seeking.” We use it inthiscontext
to describe the net force on the right hand side of Eq. 2 which acts toward the center
of the circular path of the object.
The words “centripetal” and “centrifugal” are adjectives used to describe real forces. If
you ever find yourself relying on the terms “centripetal force” or “centrifugal force,” ask
yourself which actual force you are describing (gravity, tension, friction, e.g.). If you don’t
have an answer, something has gone wrong.

Experiment
The apparatus for this experiment allows you to measure all of the important characteristics
of an object moving on a circular path at constant speed.
1. Level the apparatus with a bubble level by adjusting the heights of the legs.
2. Disconnect the spring from the mass and position the vertical metal pointer at the
horizontal position at which the mass hangs at rest. Measure the distance of the mass
(and the pointer) from the center of rotation of the apparatus. This is the radius of
the circular path on which you would like the mass to travel during the experiment.
3. Reconnect the spring, and use the string, pulley and mass hanger to measure the force
exerted by the spring on the mass when it is at the position of the pointer. Think
carefully about how to assign an uncertainty to this measurement. Hint : You may
assume the numbers stamped on the masses are accurate to within 1 gram, but the
uncertainty in the spring force is much larger than (1 g)(9.8 m/s 2 ) = 0.0098 N !
4. Measure the mass of the “mass” itself.
5. Spin the apparatus (by hand) up to the speed at which the mass hangs at the same
radial distance from the axle as the pointer. Practice maintaining this speed.
6. Now, measure the speed of the mass. You have been provided with a stopwatch for
this measurement. Hint : If you measure the time it takes for the mass to complete
several orbits, the relative uncertainty in your time measurement is smaller than if you
only time a single orbit.
7. The radius of the circular orbit is adjustable. Change the radius, adjust the counterweight
to balance the apparatus, and make another measurement. Repeat until you
have 5 measurements covering the available range of radii.
Analysis
1. Use Excel to create a graph of v2 vs. r Fcent . According to the model, the slope should
be consistent with 1 within uncertainty.
m
2. If your data appears to be compatible with a linear model, then use the linest()
function to perform a linear fit to the line (see Appendix B).
3. Add the fit line to your graph.
4. Calculate the uncertainty in one of your v2 values following the rules for the propagation
of uncertainties (see Appendix A). (Ask for help if you need
r
it.)
5. Compare the uncertainty you found in step 4 with the standard deviation in the y
estimate (σ y ) from your linear fit. Add error bars to your graph corresponding to
whichever uncertainty is larger.

Questions
Show your spreadsheet including the graph showing the best fit line and error bars on the
values to your instructor/TA and answer the following questions.
v 2
r
1. Describehowyouestablishedtheuncertaintiesinyourspring-forcemeasurements(with
the pulley and mass hanger).
2. Describe how you determined the size of the error bars on the graph – both the error
propagation and σ y approaches, and which one you chose.
3. Comment on the agreement of the model with your measurements. Was your graph
compatible with a linear model Was the slope consistent with the predicted value