Question 1
A cart on a horizontal surface is attached to a spring. The other end of the spring is attached to a wall. The cart is initially held at rest, as shown in Figure 1. When the cart is released, the system consisting of the cart and spring oscillates between the positions x = + L and x = −L. Figure 2 shows the kinetic energy of the cart-spring system as a function of the system’s potential energy. Frictional forces are negligible.
(a) On the graph of kinetic energy K versus potential energy U shown in Figure 2, the values for the x-intercept and y-intercept are the same. Briefly explain why this is true, using physics principles.
When the cart is at +L and momentarily at rest, a block is dropped onto the cart, as shown in Figure 3. The block sticks to the cart, and the block-cart-spring system continues to oscillate between −L and +L. The masses of the cart and the block are \( m_{o}\) and 3\( m_{o}\), respectively.
(b) The frequency of oscillation before the block is dropped onto the cart is \(f_{1}\). The frequency of oscillation after the block is dropped onto the cart is \(f_{2}\). Calculate the numerical value of the ratio \( \frac{f_{2}}{f_{{1}}}\).
(c) The dashed line in Figure 4 shows the kinetic energy K versus potential energy U of the block-cart-spring system after the block is dropped onto the cart. This graph is identical to the graph shown in Figure 2 for the cart-spring system before the block is dropped onto the cart.
i. Briefly explain why the two graphs must be the same, using physics principles.
ii. After the block is dropped onto the cart, consider a system that consists only of the cart and the spring. On Figure 4, sketch a solid line that shows the kinetic energy of the system that consists of the cart and the spring but not the block after the block is dropped onto the cart.
▶️Answer/Explanation
1 (a) Example Response
The maximum kinetic energy and maximum potential energy of the car-spring system are both 4 J, because energy is conserved in this system.
1(b) Example Responses
Scoring Note: Simplified versions of the above ratios also earn this point.
1(c)(i) Example Response
The maximum potential energy of the system does not depend upon the mass of the system, therefore there will be no change when the block is added.
1(c)(ii) Example Response
Question 2
2. (a) Students conduct an experiment to determine the acceleration a of a cart. The cart is released from rest at the top of the ramp at time t = 0 and moves down the ramp. The x-axis is defined to be parallel to the ramp with its origin at the top, as shown in the figure. The students collect the data shown in the following table.
i. Indicate which quantities could be graphed to yield a straight line whose slope could be used to determine the acceleration a of the cart. You may use the remaining columns in the table, as needed, to record any quantities (including units) that are not already in the table.
Vertical axis: _______________ Horizontal axis: _______________
ii. On the following grid, plot the appropriate quantities to create a graph that can be used to determine the acceleration a of the cart as it rolls down the ramp. Clearly scale and label all axes (including units), as
appropriate. Draw a straight line that best represents the data.
iii. Using the line you drew in part (a)(ii), calculate an experimental value for the acceleration a of the cart as it rolls down the ramp.
(b) The students are asked to determine an experimental value for the acceleration due to gravity \(g_{exp}\) using their data.
i. What additional quantities do the students need to measure in order to calculate \(g_{exp}\) from a ?
ii. Write an expression for the value of \(g_{exp}\) in terms of a.
(c) The students calculate the value of \(g_{exp}\) to be significantly lower than the accepted value of 9.8 \(m/s^{2}\) .
i. What is a physical reason, other than friction or air resistance, that could lead to a significant difference in the experimentally determined value of \(g_{exp}\) ?
ii. Briefly explain how the physical reason you identified in part (c)(i) would lead to the decrease in the experimentally determined value of \(g_{exp}\)
The students want to confirm that the acceleration is the same whether the cart rolls up or down the ramp. The students start the cart at the bottom and give the cart a quick push so that it rolls up the ramp and momentarily comes to rest. The x-axis is still defined to be parallel to the ramp with the origin at the top.
(d) On the following graphs, sketch the position x and velocity v as functions of time t that correspond to the scenario shown while the cart moves up the ramp.
▶️Answer/Explanation
2 (a)(i) Example Response
Vertical Axis : Position Horizontal Axis : Time squared
2 (a)(ii) Example Response
2 (a)(ii)Alternate Example Response
Scoring Note: The following tables represent the most common linearized graphs with the data that were used to determine the acceleration.
2(a)(iii) Example Response
2(b)(i) For indicating a quantity to be measured
Accept one of the following:
• The angle θ with the horizontal
• The height h and length L of the ramp
Scoring Note: Stating only the height needs to be measured can earn this point if an energy approach is used.
2(b)(ii) Example Response
2(c)(i) For identifying a physical factor that could have affected the result
Accept one of the following:
• A physical factor in the materials used (e.g., the wheels have nonnegligible rotational inertia, the ramp was bumpy, the wheels were wobbly or not perfectly round, the base of the ramp was not level, the floor was not level.)
• A physical factor in the environment (e.g., the room was being accelerated, elevator, the experiment was performed at high elevation or on a different planet.)
• A physical error in measurement collection (e.g., time, position, or angle was measured incorrectly.)
Scoring Note: A statement of “Human error” does not earn this point.
2(c)(ii) Example Response
The expression I derived for the value for \(g_{exp}\) did not take into consideration that the wheels had any rotational inertia. If the wheels have rotational inertia and are rotating, the acceleration of the cart would be less than g sinθ , so the value of \(g_{exp}\) would be less than \(9.8\frac{m}{s_{2}}\).
2(d) Example Response
Scoring Note: The following are alternate example graphs with the points the response would earn.
Question 3
3.A small block of mass \(m_{o}\) is attached to the end of a spring of spring constant \(k_{o}\) that is attached to a rod on a horizontal table. The rod is attached to a motor so that the rod can rotate at various speeds about its axis. When the rod is not rotating, the block is at rest and the spring is at its unstretched length L, as shown. All frictional forces are negligible.
(a) At time t = \(t_{1}\) , the rod is spinning such that the block moves in a circular path with a constant tangential speed \(v_{1}\) and the spring is stretched a distance \(d_{1} \) from the spring’s unstretched length, as shown in Figure . 1 At time t = \(t_{2}\), the rod is spinning such that the block moves in a circular path with a constant tangential speed \(v_{2}\) and the spring is stretched a distance \(d_{2} \) from the spring’s unstretched length, where \(d_{2}\) > \(d_{1}\) , as shown in Figure 2.
i. On the following dots, which represent the block at the locations shown in Figure 1 and Figure 2, draw the force that is exerted on the block by the spring at times t = \(t_{1}\) and t = \(t_{2}\). The spring force must be represented by a distinct arrow starting on, and pointing away from, the dot.
Note: Draw the relative lengths of the vectors to reflect the relative magnitudes of the forces exerted by the spring at both times.
ii. Referencing \(d_{1}\) and \(d_{2}\) , describe your reasoning for drawing the arrows the length that you did in part (a)(i).
iii. Is the tangential speed \(v_{1}\) of the block at time t = \(t_{1}\) greater than, less than, or equal to the tangential speed \(v_{2}\) of the block at time t = \(t_{2}\) ?
_____ \(v_{1}\) > \(v_{2}\) __X___ \(v_{1}\) < \(v_{2}\) _____ \(v_{1}\) = \(v_{2}\)
Justify your answer without using equations.
(b) Consider a scenario where the block travels in a circular path where the spring is stretched a distance d from its unstretched length L
i. Determine an expression for the magnitude of the net force \(F_{net}\) exerted on the block. Express your answer in terms of \(m_{o}\) ,\(k_{o}\) , L, d, and fundamental constants, as appropriate.
ii. Derive an equation for the tangential speed v of the block. Express your answer in terms of \(m_{o}\),\(k_{o}\) L, d, and fundamental constants, as appropriate.
(c) Does your equation for the tangential speed v of the block from part (b)(ii) agree with your reasoning from part (a) ?
____ Yes ____ No
Explain your reasoning.
▶️Answer/Explanation
3(a)(i) Example Response
3(a)(ii) Example Response
The spring force arrow drawn at \(t=t_{2}\) is longer because the spring is stretched a greater distance at that time and the spring force is related to the stretch distance.
3(a)(iii) Example Response
_____ \( v_{1}>v_{2}\) X \(v_{1}>v_{2}\) _____ \(v_{1}=v_{2}\)
The net force is the spring force. When the spring is stretched a greater length, the spring force is greater, so the net force is greater, and therefore the tangential speed is greater at \(t=t_{2}\).
3(b)(i) Example Response
\(F_{net} = \sum F = F_{s}\)
\(F_{net} = \sum F = k_{0}d\)
3(b)(ii) Example Response
3(c) Example Response
My equation from part (b)(ii) agrees with my reasoning in part (a). The tangential speed of the block as it travels in a horizontal circle is related to the distance the spring is stretched. The greater the tangential speed of the block, the greater distance the spring is stretched. The equation shows this because the d is in the numerator.
Question 4
4. A block of unknown mass is attached to a long, lightweight string that is wrapped several turns around a pulley mounted on a horizontal axis through its center, as shown. The pulley is a uniform solid disk of mass M and
radius R. The rotational inertia of the pulley is described by the equation \(I=\frac{1}{2}MR^{^{2}} \) . The pulley can rotate about its center with negligible friction. The string does not slip on the pulley as the block falls. When the block is released from rest and as the block travels toward the ground, the magnitude of the tension exerted on the block by the string is \(F_{T}\) .
(a) Determine an expression for the magnitude of the angular acceleration \(a_{D}\) of the disk as the block travels downward. Express your answer in terms of M , R, \(F_{T}\) , and physical constants as appropriate.
Scenarios 1 and 2 show two different pulleys. In Scenario 1, the pulley is the same solid disk referenced in part (a). In Scenario 2, the pulley is a hoop that has the same mass M and radius R as the disk. Each pulley
has a lightweight string wrapped around it several turns and is mounted on a horizontal axle, as shown. Each pulley is free to rotate about its center with negligible friction. In both scenarios, the pulleys begin at rest. Then both strings are pulled with the same constant force \(F_{A}\) for the same time interval Δt, causing the pulleys to rotate without the string slipping. After time interval Δt, the change in angular momentum of the disk is equal to the change in angular momentum of the hoop, but the change in rotational kinetic energy for the disk is greater than that of the hoop.
(b) Consider scenarios 1 and 2 at the end of time interval Δt. In a clear, coherent paragraph-length response that may also contain equations and drawings, explain why the change in angular momentum of both
pulleys is the same but the change in rotational kinetic energy is greater for the disk.
▶️Answer/Explanation
4(a)Example Response
\(\alpha _{D} = \frac{RF_{T}}{\frac{1}{2}MR^{2}} OR \alpha _{D} =\frac{2F_{T}}{MR}\)
4(b) Example Response
The rotational inertia, I , of the hoop is larger than the rotational inertia of the disk because the hoop’s mass is all on the outside instead of distributed throughout like the disk. Equal forces are applied to both pulleys at the same distance, which means that the torques exerted on the pulleys will also be equal. Since the same torque is applied to both pulleys for the same time period, the change in angular momentum will be the same for the disk and hoop. The magnitude of the angular velocity for the hoop will be smaller than that of the disk since angular velocity is inversely proportional to the rotational inertia \(\left ( \omega =\frac{L}{I} \right )\).
Since kinetic energy is proportional to rotational inertia and the square of angular velocity\(\left ( K_{R}=\frac{1}{2}I\omega ^{2} \right )\) , the difference in angular velocity more greatly affects the rotational kinetic energy. That means the disk will have a greater rotational kinetic energy than the hoop
Question 5
5. A rod with a sphere attached to the end is connected to a horizontal mounted axle and carefully balanced so that it rests in a position vertically upward from the axle. The center of mass of the rod-sphere system is indicated with a \(\bigotimes \) , as shown in Figure 1. The sphere is lightly tapped, and the rod-sphere system rotates clockwise with negligible friction about the axle due to the gravitational force. A student takes a video of the rod rotating from the vertically upward position to the vertically downward position. Figure 2 shows five frames (still shots) that the student selected from the video.
Note: these frames are not equally spaced apart in time.
(a) Use the frames of the video shown in Figure 2 to answer the following questions.
i. In which frame is the angular acceleration of the rod-sphere system the greatest? Justify your answer.
ii. In which frame is the rotational kinetic energy of the rod-sphere system the greatest? Briefly justify your answer.
(b) The rod-sphere system has mass M and length L, and the center of mass is located a distance \(\frac{3}{4}L\) from the axle, shown in Figure 3.
i. Derive an expression for the change in kinetic energy of the rod-sphere-Earth system from the moment shown in Frame A to the moment shown in Frame E. Express your answer in terms of M , L, and
fundamental constants, as appropriate.
ii. Briefly explain why the rod and sphere gain kinetic energy, even if Earth is not included in the system.
▶️Answer/Explanation
5(a)(i) Example Response
The angular acceleration is greatest in Frame C because angular acceleration is proportional to torque, and in Frame C the gravitational force vector is directed perpendicular to the rod (lever arm) which means this is where the torque will be the greatest.
5(a) (ii) Example Response
The rotational kinetic energy is greatest in Frame E because this is where the rod-sphere system has the greatest rotational speed since the torque has been in the same direction as the motion the entire time.
5(b) (i) Example Response
5(b)(ii) Example Response
The rod and sphere gain kinetic energy due to the positive work done by the gravitational force, which is an external force for the rod-sphere system.