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AP Physics C: Mechanics Unit 7: Gravitation

12 questions

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Unit 7: Gravitation

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Question 1

Apply energy conservation to derive the orbital speed of a satellite in a circular orbit of radius $r$ around a planet of mass $M$ . What is $v$ ?

v = \frac{2 π r}{T}

onlyB.

v = \frac{GM}{2 r}

v = \frac{GM}{r}

v = \frac{GM}{r ^{2}}

Question 2

A satellite in circular orbit of radius $r$ around Earth (mass $M_{E}$ ) has total mechanical energy $E$ . Which expression is correct?

E = - \frac{G M _{E} m}{r}

E = \frac{G M _{E} m}{2 r}

E = - \frac{G M _{E} m}{2 r}

E = 0

Question 3

Derive the gravitational potential energy by integrating the gravitational force. Starting from $F (r) = - \frac{GM m}{r ^{2}}$ (toward $M$ ), which integral correctly gives $U (r)$ with $U (\infty) = 0$ ?

U (r) = \int_{r}^{\infty} \frac{GM m}{r ^{'2}} d r^{'}

U (r) = - \int_{r}^{\infty} \frac{GM m}{r ^{'2}} d r^{'}

U (r) = \int_{0}^{r} \frac{GM m}{r ^{'2}} d r^{'}

U (r) = \int_{\infty}^{r} \frac{GM m}{r ^{'2}} d r^{'}

Question 4

Two point masses $M$ and $m$ are separated by distance $r$ . Which expression gives the magnitude of the gravitational force between them?

\frac{GM m}{r}

\frac{GM m}{r ^{2}}

\frac{GM}{r ^{2}}

GM m r^{2}

Question 5

Using Kepler's third law $T^{2} \propto r^{3}$ , if Earth's orbital radius is $1$ AU and period is $1$ yr, what is the orbital period of a planet at $4$ AU?

4

yrB.

8

yrC.

16

yrD.

64

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Question 6

The gravitational potential inside a uniform solid sphere of mass $M$ and radius $R$ at distance $r < R$ from the center is $Φ (r) = - \frac{GM}{2 R ^{3}} (3 R^{2} - r^{2})$ . What is the gravitational field $g (r)$ at $r < R$ ?

g = - \frac{GM}{R ^{3}} r

(toward center)B.

g = - \frac{GM}{R ^{2}}

(constant)C.

g = - \frac{GM}{r ^{2}}

(toward center)D.

g = 0

Question 7

A satellite of mass $m$ orbits Earth (mass $M_{E}$ , radius $R_{E}$ ) at altitude $h$ above the surface in a circular orbit. What is the orbital speed?

\frac{G M _{E}}{R _{E}}

\frac{G M _{E}}{R _{E} + h}

\frac{G M _{E}}{R _{E} + h}

\frac{G M _{E} m}{R _{E} + h}

Question 8

Using the gravitational field $g = - \frac{GM}{r ^{2}} \overset{r}{^}$ , find the work done by gravity on mass $m$ moving from $r_{1} = 2 R_{E}$ to $r_{2} = 4 R_{E}$ .

W = \frac{G M _{E} m}{4 R _{E}}

W = - \frac{G M _{E} m}{4 R _{E}}

W = \frac{G M _{E} m}{2 R _{E}}

W = 0

Question 9

A rocket launches from Earth's surface (mass $M_{E}$ , radius $R_{E}$ ). What is the minimum launch speed needed to escape Earth's gravity entirely?

\frac{G M _{E}}{R _{E}}

\frac{2 G M _{E}}{R _{E}}

\frac{2 G M _{E}}{R _{E}}

\frac{G M _{E}}{2 R _{E}}

Question 10

The gravitational potential energy of mass $m$ at distance $r$ from mass $M$ is $U (r) = - \frac{GM m}{r}$ . Why is $U$ negative?

A.Because

G

M

, and

m

are all negative quantities.B.Because we define

U = 0

at infinity, and work must be done on the system to separate the masses.C.Because the force is repulsive at close range.D.Because kinetic energy is always greater than potential energy.

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Question 11

Derive Kepler's third law ( $T^{2} \propto r^{3}$ ) for a circular orbit from Newton's law of gravitation and the definition of circular motion.

T^{2} = \frac{4 π ^{2} r ^{3}}{GM}

T^{2} = \frac{2 π r ^{3}}{GM}

T^{2} = \frac{4 π ^{2} r ^{2}}{GM}

T = \frac{2 π r}{GM / r}

Question 12

A satellite is transferred from a circular orbit of radius $r_{1}$ to a circular orbit of radius $r_{2} > r_{1}$ via a Hohmann transfer (half-ellipse). During the transfer, which statement about speed is correct at the outer transfer point?

A.The satellite must decelerate to enter the ellipse and decelerate again to circularize at

r_{2}

.B.The satellite accelerates at

r_{1}

, coasts on the ellipse, and decelerates at

r_{2}

to reach the circular orbit speed.C.The satellite accelerates at

r_{1}

, coasts on the ellipse, then accelerates again at

r_{2}

to match the circular orbit speed.D.The satellite's speed at

r_{2}

on the ellipse equals the circular orbit speed at

r_{2}

, so no second burn is needed.

AP Physics C: Mechanics Unit 7: Gravitation — Free with Answer Explanations | Test Practice Hub