Q: A 70 kg astronaut is standing on the surface of Mars, which has a mass of $$6.42 \times 10$$^{23}$$ kg and a radius of 3.39$$ \times $$10^6 m. $$What is the astronaut's weight on Mars? (Use $$G = 6.67 \times 10$$^{-11}$$ N·m^2$/kg^2$.)

$$w = 70 \times 3.71 = 259.7 N$$

Q: A 5 kg object is attached to a string and whirled in a horizontal circle of radius 2 m at a speed of 6 m/s. What is the tension in the string?

$$T = \dfrac{mv^2$}{r} = \dfrac{5 \times 36}{2} = 90 N$$

Q: Two objects, each of mass 10 kg, are placed 2 m apart. What is the magnitude of the gravitational force between them? ($$G = 6.67 \times 10$$^{-11}$$ N·m^2$/kg^2$)

$$F = \dfrac{Gm_1m_2}{r^2$} = \dfrac{6.67 \times 10$$^{-11}$$ \times 10 \times 10}{4} = 1.67 \times 10$$^{-9}$$ N

Q: A roller coaster car of mass 500 kg moves at 15 m/s over the top of a hill of radius 10 m. What is the normal force exerted on the car at the top, assuming $$g = 9.8 m/s$^2$$?

$$n = mg - \dfrac{mv^2$}{r} = 500 \times 9.8 - \dfrac{500 \times 225}{10} = 4900 - 11250 = -6350 N$\ ($$the car would lose contact$$)$$

Q: A satellite is in a low Earth orbit where $$g = 9.8 m/s$^2$$ and the radius of the orbit is $$6.4 \times 10^6$ m. What is its orbital speed?

$$v = \sqrt{gr} = \sqrt{9.8 \times 6.4 \times 10^6$} \approx 7.9 \times 10^3$ m/s

Question 1

A 70 kg astronaut is standing on the surface of Mars, which has a mass of $$6.42 	imes 10$$^{23}$$ kg and a radius of 3.39$$ 	imes $$10^6 m. $$What is the astronaut's weight on Mars? (Use $$G = 6.67 	imes 10$$^{-11}$$ N·m^2$/kg^2$.)

Accepted Answer

$$w = 70 	imes 3.71 = 259.7 N$$

Question 2

Which of the following correctly expresses the acceleration due to gravity at the surface of a planet with mass $$M$$ and radius $$R$$?

Accepted Answer

$$g = \dfrac{GM}{R^2$}$$

Question 3

A satellite is in a circular orbit of radius $$r$$ around Earth (mass $$M). $$What is the correct expression for its orbital speed $$v$$?

Accepted Answer

$$v = \sqrt{\dfrac{GM}{r}}$$

Question 4

A 1000 kg satellite orbits Earth at a radius of $$7.0 	imes 10^6$ m. What is the magnitude of the gravitational force acting on it? (Earth's mass M = 5.97$$ 	imes $$10^{24} kg$$, $$G = 6.67 	imes 10$$^{-11}$$ N·m^2$/kg^2$)

Accepted Answer

$$F = 8.13 	imes 10^3$ N

Question 5

Which of the following statements about weightlessness experienced by astronauts in orbit is correct?

Accepted Answer

Astronauts feel weightless because they are in free fall, experiencing only gravity as the net force.

Question 6

A car moves around a circular track of radius 50 m at a constant speed of 20 m/s. What is the magnitude of its centripetal acceleration?

Accepted Answer

$$a = \dfrac{v^2$}{r} = \dfrac{(20)^2$}{50} = 8 m/s$^2$$

Question 7

If the period $$T of a $$satellite's orbit doubles, by what factor does the radius $$r of $$its orbit change? (Assume $$T^2$$ \propto $$r^3.)$$

Accepted Answer

The radius increases by a factor of $$2^{2/3}$$ \approx 1.59$$

Question 8

A 5 kg object is attached to a string and whirled in a horizontal circle of radius 2 m at a speed of 6 m/s. What is the tension in the string?

Accepted Answer

$$T = \dfrac{mv^2$}{r} = \dfrac{5 	imes 36}{2} = 90 N$$

Question 9

Which of the following best describes the relationship between the period $$T$$ and the frequency $$f of an $$object in uniform circular motion?

Accepted Answer

$$f = \dfrac{1}{T}$$

Question 10

A geostationary satellite orbits Earth with a period of 24 hours. If Earth's mass is $$5.97 	imes 10$$^{24}$$ kg, what is the approximate radius of its orbit? (Use G = 6.67$$ 	imes $$10^{-11} N$$·m$$^2/kg$^2$$ and $$T = 86400 s.)$$

Accepted Answer

$$r = \left( \dfrac{$$GMT^2$$}{4\pi^2$} \$$right)^{1/3}$$ \approx 4.2 	imes 10^7$ m

Question 11

Two objects, each of mass 10 kg, are placed 2 m apart. What is the magnitude of the gravitational force between them? ($$G = 6.67 	imes 10$$^{-11}$$ N·m^2$/kg^2$)

Accepted Answer

$$F = \dfrac{Gm_1m_2}{r^2$} = \dfrac{6.67 	imes 10$$^{-11}$$ 	imes 10 	imes 10}{4} = 1.67 	imes 10$$^{-9}$$ N

Question 12

A roller coaster car of mass 500 kg moves at 15 m/s over the top of a hill of radius 10 m. What is the normal force exerted on the car at the top, assuming $$g = 9.8 m/s$^2$$?

Accepted Answer

$$n = mg - \dfrac{mv^2$}{r} = 500 	imes 9.8 - \dfrac{500 	imes 225}{10} = 4900 - 11250 = -6350 N$\ ($$the car would lose contact$$)$$

Question 13

Which of the following is a correct statement about Newton's law of universal gravitation?

Accepted Answer

The gravitational force between two masses is proportional to the product of their masses and inversely proportional to the square of the distance between them.

Question 14

A satellite is in a low Earth orbit where $$g = 9.8 m/s$^2$$ and the radius of the orbit is $$6.4 	imes 10^6$ m. What is its orbital speed?

Accepted Answer

$$v = \sqrt{gr} = \sqrt{9.8 	imes 6.4 	imes 10^6$} \approx 7.9 	imes 10^3$ m/s

Question 15

If the mass of a planet doubles and its radius remains the same, what happens to the acceleration due to gravity at its surface?

Accepted Answer

It doubles.

Question 16

A 2 kg object moves in a circle of radius 0.5 m at a constant speed of 4 m/s. What is the net force acting on the object?

Accepted Answer

$$F = \dfrac{mv^2$}{r} = \dfrac{2 	imes 16}{0.5} = 64 N$$

Question 17

Which of the following best explains why the direction of velocity changes in uniform circular motion even though the speed is constant?

Accepted Answer

The object is constantly accelerating toward the center of the circle due to centripetal acceleration.

Question 18

A satellite orbits a planet of mass $$M at a $$distance $$r$$ from its center. If the satellite's orbital radius is quadrupled, how does its orbital speed change?

Accepted Answer

It is halved.

Circular Motion, Orbits, and Gravity: Key Concepts and Applications

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