Students want to verify that the centripetal acceleration of an object undergoing uniform circular motion with tangential speed v and radius r can be described by the equation ac=v^2/r . In the experiment, the students use a variable speed turntable that can rotate about a pole at its center. A track is attached to the top of the turntable such that a cart of mass m can freely move toward and away from the center pole of the turntable. One end of a force probe is attached to the pole, and the other end is attached to the cart, as shown above. All frictional forces are negligible.
(a) The dot below represents the cart while undergoing uniform circular motion and at the location shown in the figure above. On the dot, draw and label the forces (not components) exerted on the cart. Each force must be represented by a distinct arrow starting on, and pointing away from, the appropriate dot .
Note: Draw the relative lengths of all vectors to reflect the relative magnitudes of all the forces.
(b) Using the experimental apparatus shown above, design an experimental procedure the students could use to verify that the centripetal acceleration of an object undergoing uniform circular motion with tangential speed v and radius r can be described by the equation ac=v^2/r.
i. In the table below, list the quantities and associated symbols that would be measured in the experiment and the equipment used to measure them. Also list the equipment that would be used to measure each quantity. You do not need to fill in every row. If you need additional rows, you may add them to the space just below the table.
Quantity to Be Measured
Symbol for Quantity
Equipment for Measurement
ii. Describe an experimental procedure to verify that the centripetal acceleration of an object undergoing uniform circular motion with tangential speed v and radius r can be described by the equation ac=v^2/r. Give enough detail so that another student could replicate the experiment. As needed, include a diagram of the experimental setup. Assume equipment usually found in a school physics laboratory is available.
iii. State how the measured quantities could be analyzed to verify that the centripetal acceleration of an object undergoing uniform circular motion with tangential speed v and radius r can be described by the equation ac=v^2/r.
(c) In a separate experiment, the students take force probe readings for different turntable speeds, and also measure the time it takes the turntable to make one revolution at each speed. The table shows the results of the students’ measurements. The cart’s center of mass moves in a circle of radius R=0.10 m.
Period
(s)
Force
(N)
0.75
3.5
1.25
1.3
2.0
0.49
2.5
0.32
Indicate two quantities that could be graphed to yield a straight line whose slope could be used to calculate a numerical value for the mass of the cart.
Vertical axis _______________
Horizontal axis _______________
Use the remaining columns in the table, as needed, to record any quantities that you indicated that are not given. Label each column you use and include units.
On the grid, plot the data points you indicated. Clearly scale and label both axes, including units as appropriate.
