AP Physics 1 2.9 Circular Motion Overview

Connect circular paths and orbits to centripetal acceleration, inward net force, tangential speed, radius, and Kepler’s third law.

[Maximum number: 3]

(7 points, suggested time 13 minutes)
A block of mass M is released from rest at Point A, a height 6 R above the horizontal. After being released, the block slides down a track, as shown. When released from Point A, the block does not lose contact with the track at any point. Points B and C are located at the highest points of their respective circular loops, both of radius R. All frictional forces are negligible.

Figure for Question 2.9 Circular Motion question 1 — AP Physics 1: Algebra-Based

Diagram A shows an energy bar chart that represents the gravitational potential energy $U_{g}$ of the block-Earth system and the kinetic energy K of the block at Point A, when the block is released from rest at height 6 R.

(a)

On the following dot that represents the block, draw and label the forces (not components) that are exerted on the block at the instant the block slides through Point C. Each force must be represented by a distinct arrow starting on, and pointing away from, the dot.

[ 2 ]

(i) For drawing a downward arrow labeled as the gravitational force 1 point
For drawing a downward arrow labeled as the normal force 1 point
Example Response

Figure for Answer to Question (a) — AP Physics 1: Algebra-Based

Scoring Note: Examples of appropriate labels for the gravitational force include $F_{G}, F_{g}$ , $F_{\text {grav }}, W, m g, M g$ , "grav force," " F Earth on block," " F on block by Earth," $F_{\text {Earth on Block }}, F_{\mathrm{E}, \text { Block }}$ , or $F_{\text {Block, } \mathrm{E}}$ . The labels G or g are not appropriate labels for the gravitational force.
Scoring Note: Examples of appropriate labels for the normal force include $F_{n}, F_{N}, N$ , "normal force," or "track force."
Scoring Note: Arrows of any nonzero magnitude can earn these points.

(b)

A student claims that 4 R is the minimum height of Point A, such that the block can slide through Point C without losing contact with the track after the block is released from rest. Briefly explain why this claim is incorrect.

Figure 1

[ 1 ]

[Maximum number: 5]

Figure for Question 2.9 Circular Motion question 2 — AP Physics 1: Algebra-Based

(7 points, suggested time 13 minutes)

A spacecraft of mass m is in a clockwise circular orbit of radius R around Earth, as shown in the figure above. The mass of Earth is $M_{E}$ .

(a)

Derive an equation for the orbital period T of the spacecraft in terms of $m, M_{E}, R$ , and physical constants, as appropriate. If you need to draw anything other than what you have shown in part (a) to assist in your solution, use the space below. Do NOT add anything to the figure in part (a).

[ 4 ]

(i)

Derive an equation for the orbital period T of the spacecraft in terms of $m, M_{E}, R$ , and physical constants, as appropriate. If you need to draw anything other than what you have shown in part (a) to assist in your solution, use the space below. Do NOT add anything to the figure in part (a).

[ 3 ]

For using (or implying) Newton's second law and equating the centripetal force to the gravitational force:
$F_{g}=m a=\frac{m v^{2}}{R} \quad \frac{G m M_{E}}{R^{2}}=\frac{m v^{2}}{R}$

1 point

For explicitly or implicitly determining that the speed of the spacecraft is:
$v=\frac{2 \pi R}{T}$

1 point

For a correct answer algebraically equivalent to:
$T=\sqrt{\frac{4 \pi^{2} R^{3}}{G M_{E}}}$

1 point

Note: It is acceptable to leave answer in terms of $T^{2}$ $T^{2}=\frac{4 \pi^{2} R^{3}}{G M_{E}}$

(ii)

A second spacecraft of mass 2 m is placed in a circular orbit with the same radius R. Is the orbital period of the second spacecraft greater than, less than, or equal to the orbital period of the first spacecraft? Greater than Less than Equal to
Briefly explain your reasoning.

[ 1 ]

(b)

The first spacecraft is moved into a new circular orbit that has a radius greater than R, as shown in the figure below.

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

Note: Figure not drawn to scale.
Is the speed of the spacecraft in the new orbit greater than, less than, or equal to the original speed? Greater than Less than Equal to
Briefly explain your reasoning.

[ 1 ]

LO / SP: 2.B.1.1 / 2.2; 3.A.1.1 / 1.5, 2.2; 3.B.1.3 / 1.5, 2.2; 3.B.2.1 / 1.1, 1.4, 2.2; 3.C.1.2 / 2.2 1 point
The first spacecraft is moved into a new circular orbit that has a radius greater than R, as shown in the figure below.

Note: Figure not drawn to scale.

Is the speed of the spacecraft in the new orbit greater than, less than, or equal to the original speed? Greater than Less than Equal to

Briefly explain your reasoning.

Correct answer: "Less than"

Note: If the wrong selection is made, the explanation is not graded.

For a correct explanation of why speed decreases with increasing orbital radius

1 point

Example:

Derivation step in (b)(i) shows that speed decreases with increasing R.

Learning Objectives (LO)
LO 2.B.1.1: The student is able to apply F=m g to calculate the gravitational force on an object with mass m in a gravitational field of strength g in the context of the effects of a net force on objects and systems. [See Science Practices 2.2 and 7.2]

LO 2.B.2.1: The student is able to apply $g=G M / r^{2}$ to calculate the gravitational field due to an object with mass M, where the field is a vector directed toward the center of the object of mass M. [See Science Practice 2.2]

LO 3.A.1.1: The student is able to express the motion of an object using narrative, mathematical, and graphical representations. [See Science Practices 1.5, 2.1, and 2.2]

LO 3.A.2.1: The student is able to represent forces in diagrams or mathematically using appropriately labeled vectors with magnitude, direction, and units during the analysis of a situation. [See Science Practice 1.1]

LO 3.B.1.3: The student is able to re-express a free-body diagram representation into a mathematical representation and solve the mathematical representation for the acceleration of the object. [See Science Practices 1.5 and 2.2]

LO 3.B.2.1: The student is able to create and use free-body diagrams to analyze physical situations to solve problems with motion qualitatively and quantitatively. [See Science Practices 1.1, 1.4, and 2.2]

LO 3.C.1.2: The student is able to use Newton's law of gravitation to calculate the gravitational force between two objects and use that force in contexts involving orbital motion (for circular orbital motion only in Physics 1). [See Science Practice 2.2]

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AP Physics 1 2.9 Circular Motion Overview

Exam points

Covered syllabus points

Covered syllabus points

2.9 Circular Motion question 1

Question (a)

Question (b)

2.9 Circular Motion question 2

Question (a)

Question (i)

Question (ii)

Question (b)