Experiment 6
Centripetal Force
Objective:
The objective is to experimentally verify the formula for centripetal force.
Equipment:
A centripetal force apparatus, a set of slotted weights, a weight hanger, a stop watch, a ruler, a mass balance, and a calculator
Theory:
An object performing uniform circular motion is constantly under the action of a force that acts toward its center of rotation and has a magnitude of
where M is the mass of the object, v its liner speed as it travels along the circular path, and R its radius of rotation.
Objects tend to move along straight lines unless they are forced to do otherwise. Fig. 1 shows a small object of mass M that is connected to a string of length R and is spun in a horizontal plane at a constant linear speed v. Note that in circular motion velocity is not constant. This is because of the fact that in circular motion the direction of velocity keeps changing. The velocity vector v is always perpendicular to the radius of rotation R at any point on the circular path, as shown.
Figure 1
Procedure:
Follow your instructor's directions as how to use the centripetal force apparatus and how to twirl its rotor such that it's pointed mass (bob) keeps passing exactly over the top of the stud in order to keep a constant radius of rotation. See Fig. 2 to verify the relative position of the pointed mass and the stud.
1) Detach mass M and measure it using a laboratory mass scale.
2) Choose a radius between 16 cm to 19 cm and adjust the position of the stud accordingly. This means that the horizontal distance from the top of the stud to the centerline of the rotor must be equal to your chosen radius of rotation. Use a 30-cm clear plastic ruler for this purpose. The radius does not have to be exactly 16.0, 17.0, 18.0, or 19.0cm. Any value between 16cm and 19cm will do.
3) To measure the speed V of mass M at the designated radius R, one group member should be in charge of twirling the rod of the apparatus making sure that the pointed mass passes exactly above the stud each time. Another group member should be ready to measure the time of 20, 25, or 30 full rotations with a stop watch in hand.
When the person in charge of twirling announces the constancy of R in each rotation, the time keeper should start the stop watch whenever he/she is ready and comfortable to do so. The time keeper should simultaneously start the stopwatch and announce zero as the pointed mass is passing over the stud. The next time that the pointed mass is just passing over the stud, he should count 1, and so on. At the very last desired count, he/she should turn the watch off . The elapsed time can then be read from the stopwatch. The time keeper should do the counting loud enough for other group members to hear and check the accuracy.
4) Use the radius of the circle to calculate the circumference and multiply the result by the number of rotations in order to find the total distance traveled. Divide this total distance by the total time (that was read from stopwatch) to find the average linear speed v of the rotating mass.
V = 2πRN/t where N = # of rotations.
5) With the measured values of M, V, and R, calculate F_{c }by using the centripetal force formula:
This gives you the measured value of F_{c} because your calculation is based on the measured values of M, V, and R.
The accepted Value of F_{c }:
Your instructor will show you how to attach a string to the mass and pass it over the pulley and connect it to a weight hanger.
6) Referring to Fig. 2, stack enough slotted weights on the weight hanger to cause the same stretch in the spring as the centripetal force did during the uniform circular motion. Note that you are using mass (including the hanger's mass itself) and gravity to generate force and stretch the spring. You should find the weight of the hanging mass. That gives the accepted value of F_{c } . If the total hanging mass including weight hanger is named M_{1}, then the accepted value for F_{c} is:
F_{c} = M_{1}g
7) Calculate F_{c} = M_{1}g and use it as the accepted value in this experiment.
8) Calculate a % error on F_{c } in each case between its measured and accepted values and record it in Table 1.
9) Repeat the above steps for other cases making sure that you keep the rotation radius R between 16 and 19cm.
Figure 2
Data:
Given: g = 9.8 m/s^{2}.
Measured:
Table 1
Case | R (m) |
M (kg) |
t = total rotation time (sec.) | N Number of rotations |
v =
2πRN/t (m/s) |
Measured F_{c} = Mv^{2}/R |
Accepted F_{c}= M_{1}g |
% error |
1 | ||||||||
2 | ||||||||
3 |
Calculations:
Comparison of the results:
Write down the percent error formula used as well as the calculated percent errors.
Conclusion:
State your conclusions of the experiment.
Discussion:
Provide a discussion if necessary.
Questions:
Give two examples on objects that perform circular motion and determine the source of centripetal force for each.
Suppose you tie a rock to a string and spin it in a vertical plane. In what direction will the rock move if the string breaks exactly when the rock is passing the lowest point?
Using v = 2πRf, derive a formula for F_{c} in terms of M, R, and frequency f. Simplify the formula after you substitute for v in the centripetal force formula.