AP Physics 1 Unit 6 Rotational Energy

[Maximum number: 1]

(7 points, suggested time 13 minutes)

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 ⊗, 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.

Figure 2

Use the frames of the video shown in Figure 2 to answer the following questions.

In which frame is the rotational kinetic energy of the rod-sphere system the greatest? Briefly justify your answer.

Figure 3

[Maximum number: 1]

Figure for Question 2 — AP Physics 1: Algebra-Based

One end of a string is attached to the ceiling and the other end is attached to a small sphere traveling in a circular path, as shown above, with a constant speed of 1 m/s. The string makes an angle of 30° to the vertical. If the tension in the string is 6 N and the circumference of the circle is 1 m, how much work is done on the sphere by the string as the sphere travels through one revolution?

0 J

3 J

6 J

12 J

[Maximum number: 12]

(12 points, suggested time 25 minutes)
The left end of a rod of length d and rotational inertia I is attached to a frictionless horizontal surface by a frictionless pivot, as shown above. Point C marks the center (midpoint) of the rod. The rod is initially motionless but is free to rotate around the pivot. A student will slide a disk of mass $m_{\text {disk }}$ toward the rod with velocity $v_{0}$ perpendicular to the rod, and the disk will stick to the rod a distance x from the pivot. The student wants the rod

disk system to end up with as much angular speed as possible.

(a)

Suppose the rod is much more massive than the disk. To give the rod as much angular speed as possible, should the student make the disk hit the rod to the left of point C, at point C, or to the right of point C ? To the left of C At C To the right of C
Briefly explain your reasoning without manipulating equations.

[ 1 ]

(b)

On the Internet, a student finds the following equation for the postcollision angular speed $\omega$ of the rod in this situation: $\omega=\frac{m_{\text {disk }} x v_{0}}{I}$ . Regardless of whether this equation for angular speed is correct, does it agree with your qualitative reasoning in part (a) ? In other words, does this equation for $\omega$ have the expected dependence as reasoned in part (a) ? Yes No
Briefly explain your reasoning without deriving an equation for $\omega$ .

[ 2 ]

2 points
Correct answer: "Yes"
If "No" is selected, the explanation may still earn full credit if an incorrect selection was made in part (a).
For a selection consistent with the selection from part (a)
For indicating that the equation shows that $\omega$ increases with increasing x
Example: According to the equation, $\omega$ increases with x;
a bigger x produces a bigger angular speed. This agrees with my reasoning from part (a), where I said a bigger x creates a bigger angular speed after the collision.

(c)

Another student deriving an equation for the postcollision angular speed $\omega$ of the rod makes a mistake and comes up with $\omega=\frac{I x v_{0}}{m_{\text {disk }} d^{4}}$ . Without deriving the correct equation, how can you tell that this equation is not plausible-in other words, that it does not make physical sense? Briefly explain your reasoning.

For parts (d) and (e), do NOT assume that the rod is much more massive than the disk.

[ 3 ]

3 points
For focusing on functional dependence (instead of, for example, considering units/dimensions)
For addressing $m_{\text {disk }}, I$ , or both
For correctly concluding that the equation is wrong because of the dependence on $m_{\text {disk }}, I$ , or both
Example: If $m_{\text {disk }}$ is large, then more angular momentum will be transferred during the collision. But the equation shows the angular speed decreasing with increasing $m_{\text {disk }}$ , because it is in the denominator.

Distribution

(d)

Immediately before colliding with the rod, the disk's rotational inertia about the pivot is $m_{\text {disk }} x^{2}$ and its angular momentum with respect to the pivot is $m_{\text {disk }} v_{0} x$ . Derive an equation for the postcollision angular speed $\omega$ of the rod. Express your answer in terms of $d, m_{\text {disk }}, I, x, v_{0}$ , and physical constants, as appropriate.

[ 4 ]

4 points

For using an expression of conservation of angular momentum for the disk and rod
Note: This point is not awarded for equating angular and linear momentum.

1 point

For indicating that the initial angular momentum of the system is equal to $m_{\text {disk }} v_{0} x$

1 point

For a dimensionally correct expression for the post-collision angular momentum that
includes $I \omega$

1 point

For indicating the correct rotational inertia of the system after the collision: $I+m_{\text {disk }} x^{2}$

1 point

(e)

Consider the collision for which your equation in part (d) was derived, except now suppose the disk bounces backward off the rod instead of sticking to the rod. Is the postcollision angular speed of the rod when the disk bounces off it greater than, less than, or equal to the postcollision angular speed of the rod when the disk sticks to it? Greater than Less than Equal to
Briefly explain your reasoning.

Figure for Question (e) — AP Physics 1: Algebra-Based

[ 2 ]

AP Physics 1: Algebra-Based 6 Energy and Momentum of Rotating Systems Questions

Exam points

Covered syllabus points

Covered syllabus points

Question 1

Question 2

Question 3

Question (a)

Question (b)

Question (c)

Question (d)

Question (e)