D6-1
OBJECTIVE
The objective of this lab is to measure the centripetal force acting on a mass undergoing angular motion and its centripetal acceleration.
INTRODUCTION
An object moving along a circular path at constant speed v experiences uniform circular motion.
Even though the object moves with constant speed it has acceleration due to the change in the direction of the vector velocity. Since its direction is towards the centre of the circular path in a radial direction this acceleration is known by the names of centripetal acceleration or radial acceleration.
The Magnitude of this acceleration is where r is the radius of the circular path. One can also express the centripetal acceleration in terms of the angular speed ω : since , then
To the centripetal acceleration corresponds a force, the centripetal force, whose magnitude is: and whose direction is evidently towards the centre of the path. m pin
Figure 1: Circular Motion m pin
Figure 2: Static Equilibrium
M

D6-2
The apparatus for this experiment consists of a vertical shaft (which is fit into a bearing) with a horizontal cross arm attached to the top of it (see Figure 1). On one end of the cross arm there is a counterweight (which should remain fixed in place for the duration of the experiment) and on the other end there is a mass m , which is suspended from a string and is attached to the vertical shaft via a spring.
When the shaft is rotated, the mass m has a tendency to swing outward. You should observe that as you spin the shaft (by applying a torque with your fingers) the hanging mass m moves outward and stretches the spring. The faster you spin the shaft, the further and further away from the shaft the mass will move. Clearly, the spring joining the mass to the shaft resists to the outward motion of the mass.
The number of revolutions per second performed by the mass can be observed visually. The radius of rotation can be measured with the help of a pin on the base of the apparatus (see Figure 1) and whose position can be changed. To determine its value let the mass m hang free (i.e. detach the spring from it) and adjust the position of the pin so that it lies directly below the freely hanging mass. The value of the radius of rotation is given then by the distance of the pin to the centre of the vertical shaft. When you reattach the spring to the mass and set it in motion, the pin will then act as a visual indicator as to when the mass has returned to this vertical position. The radius of rotation stays constant when the angular speed is constant.
To verify the equation that governs the uniform circular motion ( ) you will measure several values for the centripetal force and corresponding centripetal acceleration of the hanging mass. From this data you will determine an experimental value of the mass m .
The centripetal force will be found by measuring the force required to stretch the spring so that the mass m hangs vertically above the pin (see Figure 2). That is, when not in motion, suitable masses
M, attached to the mass m by a string over a pulley, should cause the same extension of the spring as when the system is in motion.
The centripetal acceleration will be determined from the measured values of the radius of rotation and corresponding angular speed.
PROCEDURE
1.
Detach the mass m from the spring and string. Measure its mass and record this value. With a pair of vernier calipers measure the diameter of the vertical shaft and record this value.
2.
Place the mass back on the string. Loosen the setscrew that holds the cross arm in place and adjust the position of the cross arm so that the point at the bottom of the mass M is about 14 cm from the shaft, and then fix the cross arm in place using the setscrew. This distance will represent the smallest radius of rotation for your experiment. Move the pin so that the top of the pin is located just below the point at the bottom of hanging mass (see Figure 1). Reattach the spring to the hanging mass.

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3.
In order to measure the radius of the path that the mass will be following, measure the distance between the pin and the shaft and record this value. This distance plus the radius of the shaft
(measured in step 1) will give you the radius r , of the path when the mass is undergoing uniform circular motion directly over the pin.
4.
Practice spinning the shaft. Once you get used to it, you should be able to spin the shaft at the right speed so that the mass travels in a circular path directly above the pin. A sheet of white paper held behind the pin will probably be useful in seeing when the mass passes exactly over the pin. Once you are comfortable that you can maintain a stable angular speed, time how long it takes for the mass to complete 25 full rotations and get three separate trials. For each trial calculate the angular speed of the cross arm (in radians per second) and then take the average of the three trials. From the average value of ω calculate the centripetal acceleration of the spinning mass using , where r is the radius (in metres) of the circular path found from step 3.
5.
To measure the centripetal force ( F c
) on the mass M take the mass holder and attach a piece of string from the mass holder to hanging mass. Add weights to the mass holder until the mass m is in static equilibrium directly over the pin (see Figure 2). The force of gravity acting on the mass
( M ) of the holder plus the weights, should be equivalent to the centripetal force that was needed to keep the hanging mass moving in a circle above the pin. Record this value as your centripetal force.
6.
Repeat steps 2-5 above using 4 other larger radii. Altogether, you should have five different values for the radius and corresponding values for centripetal accelerations and centripetal forces.
7.
If you have not already done so, plot a graph of F c
vs. a c
(you can use Maple or graph paper) and find the slope of the line that best represents your data values. Compare the value of the slope of this line to the value of the hanging mass measured in step 1.

D6-4
PRELAB QUESTIONS
1.
[1.5] What is the difference between the tangential acceleration and centripetal acceleration?
2.
[1.5] How do you define angular speed?
3.
[1] Relate the period of revolution with angular velocity.
4.
[1] What is the difference between centrifugal force and centripetal force.
REPORT WRITING INSTRUCTIONS
Read the procedure on previous two pages and prepare a table to help you record all quantities needed for this experiment.
I. Procedure:
Perform the experiment as instructed in the previous pages of the lab manual and fill in your data table.
II. Conclusions:
1.
What physical quantity does the slope of the line graphed represent?
2.
How does the measured value for the mass m compare with the one obtained from the graph?
3.
How could you improve the experimental procedure so that your results would be more accurate?