Q: On a frictionless banked curve, what is the correct relationship between the banking angle $$\theta$$, the speed $$v$$, the radius $$r$$, and the acceleration due to gravity $$g$$ for a car to make the turn without relying on friction?

$$\tan \theta = \dfrac{v^2$}{rg}$$

Q: A satellite orbits Earth at a height where the acceleration due to gravity is $$6.0\ \mathrm{m/s^2$}. If $$Earth's mass is $$5.97 \times 10$$^{24}$$\ \mathrm{kg}$$ and $$G = 6.67 \times 10$$^{-11}$$\ \mathrm{Nm^2$/kg^2$$}$, what is the satellite's orbital radius from Earth's center?

$$8.39 \times 10^6$\ \mathrm{m}$$

Q: A $$1\ \mathrm{kg}$$ mass and a $$2\ \mathrm{kg}$$ mass are separated by $$0.5\ \mathrm{m}. $$What is the magnitude of the gravitational force between them? ($$G = 6.67 \times 10$$^{-11}$$\ \mathrm{Nm^2$/kg^2$$}$)

$$5.34 \times 10$$^{-11}$$\ \mathrm{N}$$

Q: If the acceleration due to gravity on a planet is $$20\ \mathrm{m/s^2$}$$ and its radius is $$2 \times 10^6$\ \mathrm{m}$$, what is the mass of the planet? ($$G = 6.67 \times 10$$^{-11}$$\ \mathrm{Nm^2$/kg^2$$}$)

$$1.20 \times 10$$^{25}$$\ \mathrm{kg}$$

Q: A car of mass $$1000\ \mathrm{kg}$$ rounds a banked curve of radius $$50\ \mathrm{m} at$$ $$20\ \mathrm{m/s}$$ with no friction. What is the required banking angle?

$$\theta = \arctan\left(\dfrac{v^2$}{rg}\right) = \arctan\left(\dfrac{400}{50 \times 9.8}\right) \approx 39.1^\circ$$

Question 1

A car travels around a circular track of radius $$50\ \mathrm{m} at a $$constant speed, completing one revolution every $$20\ \mathrm{s}. $$What is the magnitude of its centripetal acceleration?

Accepted Answer

$$0.79\ \mathrm{m/s^2$}$$

Question 2

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

Accepted Answer

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

Question 3

A $$2\ \mathrm{kg}$$ mass is attached to a string and swung in a horizontal circle of radius $$1.5\ \mathrm{m} at a $$speed of $$3\ \mathrm{m/s}. $$What is the tension in the string?

Accepted Answer

$$12\ \mathrm{N}$$

Question 4

A car is rounding a flat, unbanked curve of radius $$40\ \mathrm{m}. If $$the coefficient of static friction between the tires and the road is $$0.5$$, what is the maximum speed the car can have without skidding?

Accepted Answer

$$14\ \mathrm{m/s}$$

Question 5

On a frictionless banked curve, what is the correct relationship between the banking angle $$	heta$$, the speed $$v$$, the radius $$r$$, and the acceleration due to gravity $$g$$ for a car to make the turn without relying on friction?

Accepted Answer

$$	an 	heta = \dfrac{v^2$}{rg}$$

Question 6

A satellite orbits Earth at a height where the acceleration due to gravity is $$6.0\ \mathrm{m/s^2$}. If $$Earth's mass is $$5.97 	imes 10$$^{24}$$\ \mathrm{kg}$$ and $$G = 6.67 	imes 10$$^{-11}$$\ \mathrm{Nm^2$/kg^2$$}$, what is the satellite's orbital radius from Earth's center?

Accepted Answer

$$8.39 	imes 10^6$\ \mathrm{m}$$

Question 7

Which of the following statements about apparent weightlessness in orbit is correct?

Accepted Answer

Objects in orbit are in free fall, so they experience apparent weightlessness even though gravity is still acting on them.

Question 8

A planet orbits a star in an elliptical orbit. According to Kepler's second law, what happens to the planet's speed as it moves closer to the star?

Accepted Answer

The planet speeds up as it gets closer to the star.

Question 9

If the period of revolution of a planet around the Sun is $$8$$ years and its average distance from the Sun is $$4$$ astronomical units (AU), which of the following is consistent with Kepler's third law?

Accepted Answer

$$T^2$$ = $$s^3$$

Question 10

A $$1\ \mathrm{kg}$$ mass and a $$2\ \mathrm{kg}$$ mass are separated by $$0.5\ \mathrm{m}. $$What is the magnitude of the gravitational force between them? ($$G = 6.67 	imes 10$$^{-11}$$\ \mathrm{Nm^2$/kg^2$$}$)

Accepted Answer

$$5.34 	imes 10$$^{-11}$$\ \mathrm{N}$$

Question 11

A $$70\ \mathrm{kg}$$ astronaut in a spacecraft accelerates upward at $$3\ \mathrm{m/s^2$}. $$What is the apparent weight of the astronaut?

Accepted Answer

$$770\ \mathrm{N}$$

Question 12

Which of the following best describes the direction of centripetal acceleration for an object in uniform circular motion?

Accepted Answer

It always points toward the center of the circle.

Question 13

A satellite is in a circular orbit around Earth. If the radius of the orbit is doubled, how does the orbital speed change?

Accepted Answer

It decreases by a factor of $$\sqrt{2}.$$

Question 14

A ball is tied to a string and swung in a vertical circle. At the top of the circle, what is the minimum speed required for the ball to maintain circular motion?

Accepted Answer

$$v = \sqrt{g r}$$

Question 15

Which of the following is the correct expression for the gravitational force between two masses $$m_1$$ and $$m_2$$ separated by a distance $$r$$?

Accepted Answer

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

Question 16

A $$1\ \mathrm{kg}$$ object moves in a circle of radius $$2\ \mathrm{m} at a $$constant speed of $$4\ \mathrm{m/s}. $$What is the magnitude of the net force acting on the object?

Accepted Answer

$$8\ \mathrm{N}$$

Question 17

If the acceleration due to gravity on a planet is $$20\ \mathrm{m/s^2$}$$ and its radius is $$2 	imes 10^6$\ \mathrm{m}$$, what is the mass of the planet? ($$G = 6.67 	imes 10$$^{-11}$$\ \mathrm{Nm^2$/kg^2$$}$)

Accepted Answer

$$1.20 	imes 10$$^{25}$$\ \mathrm{kg}$$

Question 18

A car of mass $$1000\ \mathrm{kg}$$ rounds a banked curve of radius $$50\ \mathrm{m} at$$ $$20\ \mathrm{m/s}$$ with no friction. What is the required banking angle?

Accepted Answer

$$	heta = \arctan\left(\dfrac{v^2$}{rg}ight) = \arctan\left(\dfrac{400}{50 	imes 9.8}ight) \approx 39.1^\circ$$

Question 19

A planet is twice as far from the Sun as Earth. According to Kepler's third law, how long is its orbital period?

Accepted Answer

$$2.83$$ years

Circular Motion and Gravitation: Study Notes

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