PHYS111Q : Labs

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

$$a = \frac{v^2}{r}$$ (19)

and the radial component of Newton's second law takes the form

$$m\frac{v^2}{r} = F^\mathrm{cent}$$ (20)

where $F^\mathrm{cent}$ 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. 20. 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. 20.

• 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 from a latin term meaning ``center-seeking.'' We use it in this context to describe the net force on the right hand side of Eq. 20 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 rather elaborate 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 $\frac{v^2}{r}$ vs. $F^\mathrm{cent}$. According to the model, the slope should be consistent with $\frac{1}{m}$ within uncertainty.

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), fixing the y-intercept at zero.

3. Add the fit line to your graph. Remember to use a zero intercept.

4. Calculate the uncertainty in one of your $\frac{v^2}{r}$ values following the rules for the propagation of uncertainties (see Appendix A). (Ask for help if you need it.)

5. Compare the uncertainty you found in step 4 with the standard deviation in the y estimate ($\sigma _y$) from your linear fit. Add error bars to your graph corresponding to whichever uncertainty is larger.

Group Assignment (Before You Leave Lab)

Show your spreadsheet and graph to your instructor. Discuss preliminary answers to the questions below.

Individual Assignment

Hand in a hardcopy of your spreadsheet including the graph showing the best fit line and error bars on the $\frac{v^2}{r}$ values and answers to the following questions.

1. Explain why you graphed $\frac{v^2}{r}$ vs. $F^\mathrm{cent}$ instead of $F^\mathrm{cent}$ vs. $\frac{v^2}{r}$.

2. Describe how you determined the size of the error bars on the graph - both the error propagation and $\sigma _y$ 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?

Copyright © 2003-2007, Lewis A. Riley

Updated Tue Nov 30 13:48:34 2004

This work is licensed under a Creative Commons License.