Q: A block slides down a ramp inclined at $$\theta$$ with friction coefficient $$\mu_k. $$Which of the following correctly expresses the acceleration of the block?

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

Question 1

Two crates, $$m_1 = 63$$\ \mathrm{kg}$$ and m_2$ = 128\ \mathrm{kg}$$, are in contact and at rest on a horizontal surface. A $$690\ \mathrm{N}$$ force is exerted on the $$63\ \mathrm{kg}$$ crate. If the coefficient of kinetic friction is $$0.24$$, what is the acceleration of the system?

Accepted Answer

$$a = 1.3\ \mathrm{m/s^2$}$$

Question 2

A crate of mass $$m$$ slides down a plane inclined at $$	heta = 29^\circ to $$the horizontal, with $$\mu_k = 0.17. If $$the crate starts from rest at a height of $$8.15\ \mathrm{m}$$, what will be its speed at the bottom of the incline?

Accepted Answer

$$v = 11\ \mathrm{m/s}$$

Question 3

A crate is given an initial speed of $$2.9\ \mathrm{m/s} up a$$ $$28^\circ$$ plane with $$\mu_k = 0.13. $$How far up the plane will it go before coming to rest?

Accepted Answer

$$d = 0.53\ \mathrm{m}$$

Question 4

Suppose the coefficient of kinetic friction between $$m_A$$ and the plane is $$\mu_k = 0.15$$, and $$m_A = m_B = 2.7\ \mathrm{kg}. If$$ $$	heta = 34^\circ$$, what is the magnitude of the acceleration of $$m_A$$ and $$m_B as$$ $$m_B$$ moves down?

Accepted Answer

$$a = 1.6\ \mathrm{m/s^2$}$$

Question 5

Two blocks, $$m_A = m_B = 5.2\ \mathrm{kg}$$, slide down a ramp inclined at $$	heta = 40^\circ. $$The coefficients of friction are $$\mu_A = 0.22$$ and $$\mu_B = 0.35. $$What is the acceleration of the blocks?

Accepted Answer

$$a = 4.2\ \mathrm{m/s^2$}$$

Question 6

A $$1200\ \mathrm{kg}$$ car rounds a turn of radius $$60.0\ \mathrm{m} on a $$flat road. If the coefficient of friction between tires and road is $$0.80$$, what is the maximum speed with which the car can round the turn?

Accepted Answer

$$v = 22\ \mathrm{m/s}$$

Question 7

Is the maximum speed at which a car can round a flat turn (limited by friction) dependent on the mass of the car?

Accepted Answer

The result is independent of the car's mass.

Question 8

At what minimum speed must a roller coaster be traveling when upside down at the top of a circle so that the passengers do not fall out? Assume the radius of the circle is $$8.6\ \mathrm{m}.$$

Accepted Answer

$$v_{min} = 8.6\ \mathrm{m/s}$$

Question 9

A jet pilot takes his aircraft in a vertical loop. If the jet is moving at a speed of $$1150\ \mathrm{km/h} at $$the lowest point of the loop, what is the minimum radius of the circle so that the normal force on the pilot does not exceed $$6100\ \mathrm{N}$$?

Accepted Answer

This cannot be determined without the pilot's mass.

Question 10

An object moves in a circle of radius $$21\ \mathrm{m}$$ with its speed given by $$v = 3.6 + 1.5t^2$, where v$ is in meters per second and t$ in seconds. At t = 3.5$$\ \mathrm{s}$$, what is the tangential acceleration?

Accepted Answer

$$a_{tan} = 11\ \mathrm{m/s^2$}$$

Question 11

A ball of mass $$0.65\ \mathrm{kg} is $$attached to a vertical post by strings of length $$L_1 = 1.1$$\ \mathrm{m}$$ and L_2$ = 1.7\ \mathrm{m}$$, and set moving in a horizontal circle. What is the maximum speed for the lower string to be slack (tension in cord $$= 0$$)?

Accepted Answer

$$v = 3.0\ \mathrm{m/s}$$

Question 12

If the ball in the previous question is moving at twice the speed found in part A, what is the tension in the $$L_2$$ string?

Accepted Answer

$$F_{T2} = 8.4\ \mathrm{N}$$

Question 13

A block slides down a ramp inclined at $$	heta$$ with friction coefficient $$\mu_k. $$Which of the following correctly expresses the acceleration of the block?

Accepted Answer

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

Question 14

A car of mass $$m$$ rounds a curve of radius $$r at $$speed $$v. $$The frictional force required to keep the car from skidding is:

Accepted Answer

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

Question 15

A roller coaster moves in a vertical circle of radius $$R. $$What is the minimum speed at the top of the circle so that the passengers do not fall out?

Accepted Answer

$$v_{min} = \sqrt{gR}$$

Question 16

A block of mass $$m$$ slides down a frictionless incline of height $$h. $$What is its speed at the bottom?

Accepted Answer

$$v = \sqrt{2gh}$$

Question 17

A block of mass $$m is $$pulled by a force $$F on a $$horizontal surface with friction coefficient $$\mu_k. $$What is the acceleration?

Accepted Answer

$$a = \frac{F - \mu_k mg}{m}$$

Question 18

A ball of mass $$m$$ moves in a horizontal circle of radius $$r at $$speed $$v. $$What is the tension in the string?

Accepted Answer

$$T = \frac{mv^2$}{r}$$

Dynamics and Circular Motion: Friction, Inclined Planes, and Centripetal Forces

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