Q: A 10 kg car drives over a round hump with a radius of 10 m. What is the maximum speed the car can have at the top of the hump without losing contact with the ground? (Take $$g = 9.8\ \mathrm{m/s^2$})$$

$$v = \sqrt{gR} = \sqrt{9.8 \times 10} = 9.9\ \mathrm{m/s}$$

Q: A block of mass $$m_2$$ rests on an inclined plane of angle $$\theta$$ and is connected by a string over a pulley to a hanging mass $$m_1. $$The coefficient of kinetic friction between $$m_2$$ and the plane is $$\mu_k. $$Which equation gives the condition for $$m_1 to $$pull $$m_2 up $$the incline?

$$m_1g \u003e m_2g\sin\theta + \mu_k m_2g\cos\theta$$

Q: A ball of mass $$1\ \mathrm{kg} is $$attached to a rope of length $$0.5\ \mathrm{m}$$ and rotates in a horizontal circle with a period of $$1\ \mathrm{s}. $$What is the magnitude of the tension in the rope?

$$T = mg + m \$$omega^2$$ r = 1 \times 9.8 + 1 \times (2\pi)^2$ \times 0.25 = 9.8 + 9.87 = 19.67\ \mathrm{N}$$

Question 1

A block of mass $$m is $$placed on an inclined plane with angle $$\alpha to $$the horizontal. If the coefficient of static friction is $$\mu_s$$, what is the minimum value of $$\mu_s$$ required to prevent the block from sliding down the plane?

Accepted Answer

$$\mu_s = 	an \alpha$$

Question 2

A 10 kg car drives over a round hump with a radius of 10 m. What is the maximum speed the car can have at the top of the hump without losing contact with the ground? (Take $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$v = \sqrt{gR} = \sqrt{9.8 	imes 10} = 9.9\ \mathrm{m/s}$$

Question 3

A block of mass $$m_2$$ rests on an inclined plane of angle $$	heta$$ and is connected by a string over a pulley to a hanging mass $$m_1. $$The coefficient of kinetic friction between $$m_2$$ and the plane is $$\mu_k. $$Which equation gives the condition for $$m_1 to $$pull $$m_2 up $$the incline?

Accepted Answer

$$m_1g \u003e m_2g\sin	heta + \mu_k m_2g\cos	heta$$

Question 4

A car of mass $$m is $$rounding a flat curve of radius $$R. If $$the coefficient of static friction between the tires and the road is $$\mu_s$$, what is the maximum speed the car can have without slipping?

Accepted Answer

$$v_{max} = \sqrt{\mu_s g R}$$

Question 5

For a car rounding a banked curve of radius $$R$$ and banking angle $$\beta$$, what is the formula for the speed at which no friction is required to make the turn?

Accepted Answer

$$v = \sqrt{gR 	an \beta}$$

Question 6

A conical pendulum consists of a mass $$m$$ attached to a string of length $$L$$, making an angle $$	heta$$ with the vertical as it rotates in a horizontal circle of radius $$r. $$Which equation correctly relates the tension $$T in $$the string to the forces involved?

Accepted Answer

$$T \cos 	heta = mg$$

Question 7

A ball of mass $$1\ \mathrm{kg} is $$attached to a rope of length $$0.5\ \mathrm{m}$$ and rotates in a horizontal circle with a period of $$1\ \mathrm{s}. $$What is the magnitude of the tension in the rope?

Accepted Answer

$$T = mg + m \$$omega^2$$ r = 1 	imes 9.8 + 1 	imes (2\pi)^2$ 	imes 0.25 = 9.8 + 9.87 = 19.67\ \mathrm{N}$$

Question 8

In the SpinLaunch suborbital accelerator, a payload is spun in a vacuum chamber before being released tangentially. What path does the payload follow immediately after release?

Accepted Answer

It moves in a straight line tangent to the circle at the point of release.

Question 9

A block of mass $$m is on a $$slab that is being accelerated horizontally. If the coefficient of static friction is $$\mu_s$$, what is the maximum acceleration before the block slips?

Accepted Answer

$$a_{max} = \mu_s g$$

Question 10

A satellite orbits Earth in a circular path. Which force provides the necessary centripetal acceleration?

Accepted Answer

The gravitational force between the Earth and the satellite.

Question 11

A block slides down an inclined plane with angle $$\alpha$$ and coefficient of kinetic friction $$\mu_k. $$What is the acceleration of the block?

Accepted Answer

$$a = g(\sin \alpha - \mu_k \cos \alpha)$$

Question 12

A car of mass $$m is $$rounding a flat curve of radius $$R at $$speed $$v. $$What is the direction of the frictional force acting on the car?

Accepted Answer

Toward the center of the curve.

Question 13

Which of the following best describes the net force acting on a particle in uniform circular motion?

Accepted Answer

The net force is always directed toward the center of the circle.

Question 14

A conical pendulum has a mass $$m$$ rotating in a circle of radius $$r at $$constant speed $$v. $$What is the expression for the period $$T of $$rotation?

Accepted Answer

$$T = \frac{2\pi r}{v}$$

Question 15

A block of mass $$m is at $$rest on an inclined plane with angle $$\alpha. $$What is the normal force acting on the block?

Accepted Answer

$$N = mg \cos \alpha$$

Question 16

A ball is attached to a string and swung in a horizontal circle. If the string breaks, what path does the ball follow?

Accepted Answer

It moves in a straight line tangent to the circle at the point of release.

Inclined Planes and Dynamics of Circular Motion: Study Notes for University Physics I

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