Newton’s First & Second Law in Rotational Form (College Board AP® Physics 1: Algebra-Based): Exam Questions

Figure 1

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 in Figure 1. The pulley is a uniform solid disk of mass and radius . The rotational inertia of the pulley is described by the equation . 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 .

i) Derive an expression for the magnitude of the angular acceleration of the disk as the block travels downward. Express your answer in terms of , , , and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

ii) Derive an expression for the tangential acceleration of a point located on the outer edge of the disk. Express your answer in terms of , , , and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Figure 2

Scenarios 1 and 2 show two different pulleys. In Scenario 1, the pulley is a uniform solid disk of mass and radius . In Scenario 2, the pulley is a hoop that has the same mass and radius as the disk. Each pulley has a lightweight string wrapped around it several turns and is mounted on a horizontal axle, as shown in Figure 2. 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 for the same time interval , causing the pulleys to rotate without the string slipping.

The magnitude of the angular acceleration of the disk is . The magnitude of the angular acceleration of the hoop is .

Indicate whether is greater than, less than, or equal to .

⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽

Justify your reasoning.

Figure 1

Two pulleys with different radii are attached to each other so that they rotate together about a horizontal axle through their common center. There is negligible friction in the axle. Object 1 has mass and hangs from a light string wrapped around the larger pulley of radius , while Object 2 has mass and hangs from another light string wrapped around the smaller pulley of radius , as shown in Figure 1. At time , the pulleys are released from rest and the objects begin to accelerate.

The following dots represent the objects in Figure 1. The arrow labeled represents the gravitational force exerted on Object 1. On the dots, draw and label the other forces (not components) exerted on the objects after the pulleys are released. Each force must be represented by a distinct arrow starting on and pointing away from each dot. The lengths of the arrows should reflect the relative magnitudes of the forces exerted on each object.

The rotational inertia of the two pulleys is .

Derive an expression for the angular acceleration of the pulleys after they are released. Express your answer in terms of , , , and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

At a later time , the string of Object 1 is cut while the objects are still moving and the pulley is still rotating.

Immediately after the string is cut, indicate whether the angular velocity and angular acceleration of the pulley are in the same direction or opposite directions. Briefly justify your reasoning.

Figure 1

Students are investigating balancing systems using the following setup. The students have a spring scale of negligible mass that is fixed to one end of a uniform meterstick. The center of the meterstick is attached to a stand on which the meterstick can pivot. There is a hook of negligible mass fixed to the top of a block of mass . The hook can be attached to the meterstick through one of the small holes in the meterstick, as shown in Figure 1. The students do not have a direct way to measure the mass of the block. The block cannot be attached to the spring scale.

The students are asked to take measurements that will allow them to create a linear graph whose slope could be used to determine the mass of the block.

Describe an experimental procedure to collect data that would allow the students to determine . Include any steps necessary to reduce experimental uncertainty.

Describe how the data collected in part a) could be graphed and how that graph would be analyzed to determine .

Figure 2

The students have an identical meterstick of mass that is now attached to an axle that is fixed to a wall. The meterstick is free to rotate with negligible friction about the axle. The meterstick is suspended horizontally by a string that is connected to a spring scale of negligible mass, as shown in Figure 2.

The angle that the string makes with the meterstick can be varied by attaching the string to one of the pegs located along the wall. The students use the spring scale to measure the tension required to hold the meterstick horizontal. Table 1 shows the measured values of and .

Table 1

The students correctly determine that the relationship between and is given by

The students create a graph with plotted on the horizontal axis.

i) Indicate a measured or calculated quantity that could be plotted on the vertical axis to yield a linear graph whose slope can be used to calculate an experimental value for the mass of the meterstick.

Horizontal Axis: ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ Vertical Axis: ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽

ii) On the grid in Figure 3, plot the data points for the quantities indicated in part c)i) that can be used to determine . Clearly scale and label all axes, including units, as appropriate.

Figure 3

iii) Draw a straight best-fit line for the data graphed in part c)ii).

Using the best-fit line that you drew in part c)iii), calculate an experimental value for the mass of the meterstick.

Figure 1

A nonuniform bar is attached to a wall by a fixed pivot at its left end, as shown in Figure 1. The bar has length and mass . Its mass is distributed such that the bar's center of mass is a distance from the bar's free end and the bar's rotational inertia about the pivot is . A string attached to the wall and to the bar keeps it horizontal.

The string is cut, and the bar starts rotating from its initially horizontal position. As the bar rotates, a constant frictional torque of magnitude is exerted on the bar by the pivot.

Starting with Newton’s second law in rotational form, derive an expression for the bar's angular acceleration immediately after the string is cut. Express your answer in terms of , , and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

After the string is cut, the bar rotates until it is hanging vertically along the wall.

On the axes in Figure 2, draw a graph of the bar's angular acceleration as a function of the bar's angle from the horizontal. Take the positive direction to be clockwise, the direction in which the bar rotates before hitting the wall.

Figure 2