Q: A 350 kg Mars lander has a weight of 3430 N on Earth. What is its weight on Mars? (Given $$g_{Mars} = 3.7\ \mathrm{m/s^2$})$$

$$W = 350 \times 3.7 = 1295\ \mathrm{N}$$

Q: The energy required to escape Earth's gravity for a 6000 kg spacecraft is closest to:

$$\Delta U = 3.75 \times 10$$^{11}$$\ \mathrm{J}$$

Q: Hailey's comet has a perihelion distance of $$5.26 \times 10^7$ km and an aphelion distance of 5.24$$ \times $$10^9 km. $$What is the semi-major axis $$a of $$its orbit?

$$a = \dfrac{5.26 \times 10^7$ + 5.24 \times 10^9$}{2} = 2.67 \times 10^9$\ \mathrm{km}$$

Q: A torque $$\tau is $$applied to a spinning top, causing precession. If the torque is due to gravity and the top is tilted at angle $$\theta$$, which formula gives the precession angular velocity $$\Omega$$?

$$\Omega = \dfrac{Mgr\sin\theta}{L}$$

Question 1

A 180 N ladder is 5.0 m long and rests against a wall, making a $$53.1^\circ$$ angle with the ground. If a person of mass 80 kg stands halfway up the ladder, what is the normal force exerted by the ground on the ladder? (Assume $$g = 9.8\ \mathrm{m/s^2$}$$ and the ladder is uniform.)

Accepted Answer

$$n = 980\ \mathrm{N} + 180\ \mathrm{N} = 1160\ \mathrm{N}$$

Question 2

A gyroscope has angular momentum $$\vec{L}$$ pointing along its spin axis. When a torque $$\vec{	au} is $$applied perpendicular to $$\vec{L}$$, what happens to the gyroscope?

Accepted Answer

The direction of $$\vec{L}$$ changes, but its magnitude remains constant.

Question 3

A 350 kg Mars lander has a weight of 3430 N on Earth. What is its weight on Mars? (Given $$g_{Mars} = 3.7\ \mathrm{m/s^2$})$$

Accepted Answer

$$W = 350 	imes 3.7 = 1295\ \mathrm{N}$$

Question 4

The energy required to escape Earth's gravity for a 6000 kg spacecraft is closest to:

Accepted Answer

$$\Delta U = 3.75 	imes 10$$^{11}$$\ \mathrm{J}$$

Question 5

A satellite of mass 1000 kg orbits Earth at an altitude of 300 km above the surface. If Earth's radius is $$6.37 	imes 10^6$\ \mathrm{m}$$ and $$g = 9.8\ \mathrm{m/s^2$}$$, what is the satellite's orbital speed?

Accepted Answer

$$v = 7730\ \mathrm{m/s}$$

Question 6

Which of the following best describes Kepler's First Law?

Accepted Answer

Planets move in elliptical orbits with the Sun at one focus.

Question 7

Hailey's comet has a perihelion distance of $$5.26 	imes 10^7$ km and an aphelion distance of 5.24$$ 	imes $$10^9 km. $$What is the semi-major axis $$a of $$its orbit?

Accepted Answer

$$a = \dfrac{5.26 	imes 10^7$ + 5.24 	imes 10^9$}{2} = 2.67 	imes 10^9$\ \mathrm{km}$$

Question 8

The total mechanical energy $$E of a $$satellite of mass $$m in a $$circular orbit of radius $$r$$ around Earth (mass $$M) is$$:

Accepted Answer

$$E = -\dfrac{GMm}{2r}$$

Question 9

A black hole's Schwarzschild radius $$r_s is $$given by:

Accepted Answer

$$r_s = \dfrac{2GM}{c^2$}$$

Question 10

If the escape velocity from a planet is $$v_{esc}$$, which of the following is true?

Accepted Answer

$$v_{esc} = \sqrt{\dfrac{2GM}{R}}$$

Question 11

A satellite in a circular orbit has a period $$T$$ and radius $$r. $$Which equation correctly relates $$T$$ and $$r$$?

Accepted Answer

$$T^2$$ = \dfrac{4\$$pi^2$$}{GM} $$r^3$$

Question 12

A torque $$	au is $$applied to a spinning top, causing precession. If the torque is due to gravity and the top is tilted at angle $$	heta$$, which formula gives the precession angular velocity $$\Omega$$?

Accepted Answer

$$\Omega = \dfrac{Mgr\sin	heta}{L}$$

Question 13

Which of the following statements about planetary orbits is correct according to Kepler's Second Law?

Accepted Answer

A planet sweeps out equal areas in equal times.

Question 14

If a satellite's total mechanical energy becomes zero, what happens to its orbit?

Accepted Answer

It escapes Earth's gravity and moves away on a parabolic trajectory.

Question 15

A satellite orbits at a radius $$r$$ from the center of Earth. If the radius is doubled, how does the orbital speed $$v$$ change?

Accepted Answer

$$v$$ decreases by a factor of $$\sqrt{1/2}$$

Question 16

The centripetal acceleration $$a_c$$ for a satellite in a circular orbit of radius $$r$$ and speed $$v is$$:

Accepted Answer

$$a_c = \dfrac{v^2$}{r}$$

Question 17

A satellite of mass $$m is in a $$circular orbit of radius $$r$$ around a planet of mass $$M. $$What is the gravitational force acting on the satellite?

Accepted Answer

$$F = \dfrac{GMm}{r^2$}$$

Question 18

If the eccentricity $$e of a $$planet's orbit is zero, what is the shape of the orbit?

Accepted Answer

A circle

Question 19

A satellite is launched into a circular orbit. Which of the following increases as the orbital radius increases?

Accepted Answer

Orbital period

Circular Motion, Gravitation, and Orbital Mechanics: Study Notes

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