Experiment  13
Centripetal Force

Objectives

In this experiment we will investigate the relationship between centripetal force and rotational frequency.

Introduction

When an object moves in a circular path its velocity is constantly changing in direction.  This is true regardless of whether the speed of the object is changing, or not.  The changing velocity implies that there must be an acceleration which is responsible for the continuously changing direction of the path of the object.  This acceleration is the centripetal acceleration.  Consideration of the rate at which the velocity changes direction shows that the centripetal acceleration is directed toward the center of curvature of the object’s path and has magnitude given by

                  (1)

Associated with this acceleration, by Newton’s 2nd Law, is a centripetal force.  Also directed toward the center of curvature of the object’s path, this force is given by

                   (2)

The centripetal force apparatus is such that the restoring force exerted by the spring on the rotating mass is proportional to the number of turns of the collar which determines the position of the fixed end of the spring.

                                (3)

Here Fo is the force in the spring for the initial setting of the collar, and k is a constant corresponding to the change in the spring force per turn of the collar.

Combining (2) and (3):                        

Procedure

 1)            Adjust the collar so that there is a small, but non-zero, force exerted by the spring.

 2)            Making sure the centripetal force apparatus is securely clamped in the chuck of the rotator, gradually increase the speed of rotation until the pointer is deflected up to the index mark.  Carefully adjust the speed until the pointer remains at the index mark, or oscillates about it.

 3)            Record the speed of rotation.

 4)            Stop the rotation of the apparatus and increase the tension in the spring by rotating the collar two full turns.  Once again adjust the rotation speed until the pointer remains at the index mark, or oscillates about it.

 5)            Repeat 4 for six additional (2-turn) increases of the spring tension

6)                  Plot the rotational frequency (in rotations per second) squared vs. number of turns of the collar, and do a least squares fit to your data.  The slope of the least squares fit line is

 

 

 

7)                  Remove the centripetal force apparatus from the chuck, return the collar to its original position, and suspend the apparatus from a secure support.

8)                  Determine the total amount of weight required to cause the pointer to be deflected to the index mark.  Include in this total the weight of the rotating mass.

9)                  Repeat for each of the other positions of the collar used in the rotation part of the experiment.

10)               Using a caliper, measure the radius of the rotational motion – from the axis of rotation, indicated by the mark scribed on the frame, to the index mark scribed on the rotating mass.

11)               Plot weight vs. number of turns of the collar, and do a least squares fit to your data.  The slope of this line is k’stat.

12)               Compare your two values for k’, by finding a percent deviation.