Q: A 0.5-kg ball is tied to the end of a 4-m light cord and swung in a horizontal plane at constant speed, making an angle of 30° with the vertical. What is the tension in the cord if the ball's speed is 6 m/s?

$$T = \frac{mv^2$}{r} + mg \cos \theta$$

Q: At what height above Earth is the gravitational force on a 1000-kg satellite equal to 1000 N?

$$h = 5.37 \times 10^6$ \text{ m}$$

Q: A 2000-kg spacecraft is blasting away from the surface of an unknown planet the same size as Earth. At 1500 km above the surface, an instrument onboard reads the gravitational force to be 1800 N. What is the planet's mass?

$$M = 4.3 \times 10$$^{24}$$ \text{ kg}$$

Q: A satellite orbits at an orbital period of 2 hours around the Moon. What is the satellite's orbital altitude?

$$h = 1.6 \times 10^6$ \text{ m}$$

Q: A distant planet orbits a star 3 times the mass of our Sun. This planet of mass $$6 \times 10$$^{24}$$ kg feels a gravitational force of 3.2$$ \times $$10^{22} N. $$What is the planet's orbital distance if the star's mass is $$6 \times 10$$^{30}$$ kg?

$$r = 2.5 \times 10$$^{11}$$ \text{ m}$$

Question 1

A 0.5-kg ball is tied to the end of a 4-m light cord and swung in a horizontal plane at constant speed, making an angle of 30° with the vertical. What is the tension in the cord if the ball's speed is 6 m/s?

Accepted Answer

$$T = \frac{mv^2$}{r} + mg \cos 	heta$$

Question 2

Which of the following correctly describes the direction of the centripetal force for an object moving in a circle?

Accepted Answer

Perpendicular to the velocity and toward the center

Question 3

A 1500-kg car passes over a bump in a roadway that follows the arc of a circle of radius 20 m. What force does the road exert on the car as the car moves over the top of the bump if the car moves at a constant speed of 10 m/s?

Accepted Answer

$$F = mg - \frac{mv^2$}{r}$$

Question 4

A 1000-kg car can drive around a flat curve of radius 50 m with a maximum speed of 20 m/s before slipping. What is the coefficient of static friction between the car and the road?

Accepted Answer

$$\mu_s = \frac{v^2$}{gr}$$

Question 5

A truck can go around a flat curve of radius 150 m with a maximum speed of 32 m/s before slipping. Calculate the maximum speed if it can go around a tighter curve of radius 75 m.

Accepted Answer

$$v_{max} = 16 	ext{ m/s}$$

Question 6

For a banked curve with no friction, what is the formula for the speed at which a car can take the curve without sliding?

Accepted Answer

$$v = \sqrt{gr 	an 	heta}$$

Question 7

Newton's Universal Law of Gravitation states that the gravitational force between two objects is:

Accepted Answer

$$F = G \frac{m_1$ m_2$}{r^2$}$$

Question 8

Two spheres of mass 10 kg and 20 kg are positioned 5 m apart. Suppose you place a third sphere directly in between the two spheres, so that all three are on the same line. How far from the 10 kg mass would you have to place the third sphere so that the net gravitational force on it is zero?

Accepted Answer

$$x = 3.33 	ext{ m}$$

Question 9

At what height above Earth is the gravitational force on a 1000-kg satellite equal to 1000 N?

Accepted Answer

$$h = 5.37 	imes 10^6$ 	ext{ m}$$

Question 10

A 2000-kg spacecraft is blasting away from the surface of an unknown planet the same size as Earth. At 1500 km above the surface, an instrument onboard reads the gravitational force to be 1800 N. What is the planet's mass?

Accepted Answer

$$M = 4.3 	imes 10$$^{24}$$ 	ext{ kg}$$

Question 11

The acceleration due to gravity at a distance $$r$$ from the center of a planet of mass $$M is $$given by:

Accepted Answer

$$g = G \frac{M}{r^2$}$$

Question 12

A satellite orbits at an orbital period of 2 hours around the Moon. What is the satellite's orbital altitude?

Accepted Answer

$$h = 1.6 	imes 10^6$ 	ext{ m}$$

Question 13

For a satellite in circular orbit, the orbital speed $$v is $$related to the radius $$r$$ and mass $$M of $$the planet by:

Accepted Answer

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

Question 14

Kepler's Third Law states that for any circular orbit, the orbital period squared is proportional to the orbital radius cubed. Which equation represents this law?

Accepted Answer

$$T^2$$ \propto $$r^3$$

Question 15

A black hole has a mass $$M. $$The Schwarzschild radius $$R_s is $$given by:

Accepted Answer

$$R_s = \frac{2GM}{c^2$}$$

Question 16

If the Sun were instantly replaced by an Earth-sized black hole, what would be the net acceleration of objects on Earth due to this black hole?

Accepted Answer

$$g = G \frac{M_{BH}}{R_{Earth}^2}$$

Question 17

A distant planet orbits a star 3 times the mass of our Sun. This planet of mass $$6 	imes 10$$^{24}$$ kg feels a gravitational force of 3.2$$ 	imes $$10^{22} N. $$What is the planet's orbital distance if the star's mass is $$6 	imes 10$$^{30}$$ kg?

Accepted Answer

$$r = 2.5 	imes 10$$^{11}$$ 	ext{ m}$$

Question 18

Which of the following is true about the net gravitational force on a mass at the center of an equilateral triangle formed by three equal masses at the vertices?

Accepted Answer

The net force is zero due to symmetry

Circular Motion & Gravitation: Study Notes (College Physics)

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