Q: A wet bar of soap slides freely down a ramp 9.0 m long inclined at $$8.0^\circ. $$The coefficient of kinetic friction is $$\mu_k = 0.060. $$What is the time required for the soap to reach the bottom of the ramp?

$$t = \sqrt{\dfrac{2L}{g(\sin\theta - \mu_k \cos\theta)}}$$ where $$L = 9.0 m$$, $$\theta = 8.0^\circ$$, $$\mu_k = 0.060$$

Q: Two masses, $$m_A = 2.0 kg $$and $$m_B = 5.0 kg$$, are connected by a string over two inclines ($$51^\circ$$ and $$21^\circ). $$The coefficient of kinetic friction is $$\mu_k = 0.30. If$$ $$m_A$$ moves up and $$m_B$$ moves down, what is the acceleration of the system?

$$a = \dfrac{m_B g \sin\theta_B - m_A g \sin\theta_A - \mu_k (m_A g \cos\theta_A + m_B g \cos\theta_B)}{m_A + m_B}$$

Q: A child slides down a slide with a $$34^\circ$$ incline. At the bottom, her speed is half what it would have been if the slide were frictionless. What is the coefficient of kinetic friction between the slide and the child?

$$\mu_k = \tan\theta - \dfrac{1}{2}$$

Question 1

A wet bar of soap slides freely down a ramp 9.0 m long inclined at $$8.0^\circ. $$The coefficient of kinetic friction is $$\mu_k = 0.060. $$What is the time required for the soap to reach the bottom of the ramp?

Accepted Answer

$$t = \sqrt{\dfrac{2L}{g(\sin	heta - \mu_k \cos	heta)}}$$ where $$L = 9.0 m$$, $$	heta = 8.0^\circ$$, $$\mu_k = 0.060$$

Question 2

Police investigators measure 72-m-long skid marks from a car that nearly stopped before colliding. The coefficient of kinetic friction between rubber and pavement is $$\mu_k = 0.80. $$What was the car's initial speed? (Assume a level road.)

Accepted Answer

$$v_0$$ = \sqrt{2 \mu_k g d}$$

Question 3

A block of mass $$m_A$$ sits on a table connected by a string over a pulley to a hanging mass $$m_B = 2.0 kg. $$The coefficient of static friction between $$m_A$$ and the table is $$0.40$$, and the coefficient of kinetic friction is $$0.30. $$What is the minimum value of $$m_A to $$keep the system from starting to move?

Accepted Answer

$$m_A = \dfrac{m_B}{\mu_s}$$ where $$\mu_s = 0.40$$

Question 4

Two masses, $$m_A = 2.0 kg $$and $$m_B = 5.0 kg$$, are connected by a string over two inclines ($$51^\circ$$ and $$21^\circ). $$The coefficient of kinetic friction is $$\mu_k = 0.30. If$$ $$m_A$$ moves up and $$m_B$$ moves down, what is the acceleration of the system?

Accepted Answer

$$a = \dfrac{m_B g \sin	heta_B - m_A g \sin	heta_A - \mu_k (m_A g \cos	heta_A + m_B g \cos	heta_B)}{m_A + m_B}$$

Question 5

A child slides down a slide with a $$34^\circ$$ incline. At the bottom, her speed is half what it would have been if the slide were frictionless. What is the coefficient of kinetic friction between the slide and the child?

Accepted Answer

$$\mu_k = 	an	heta - \dfrac{1}{2}$$

Question 6

How large must the coefficient of static friction be between the tires and the road if a car is to round a level curve of radius $$85 m at a $$speed of $$95 km/h$$?

Accepted Answer

$$\mu_s = \dfrac{v^2$}{r g}$$

Question 7

A space shuttle is in orbit $$400 km $$from Earth's surface, circling Earth every $$90$$ min. What is the centripetal acceleration of the shuttle in terms of $$g$$, the gravitational acceleration at Earth's surface?

Accepted Answer

$$a_c = \dfrac{4 \pi^2$ r}{T^2$}$$

Question 8

A bucket of mass $$2.00 kg is $$whirled in a vertical circle of radius $$1.10 m. At $$the lowest point, the tension in the rope is $$25.0 N. $$What is the speed of the bucket at the lowest point?

Accepted Answer

$$v = \sqrt{\dfrac{T - mg}{m} \cdot r}$$ where $$T = 25.0 N$$, $$m = 2.00 kg$$, $$r = 1.10 m$$

Question 9

A jet pilot takes his aircraft in a vertical loop at a speed of $$1200 km/h at $$the lowest point. What is the minimum radius of the circle so that the centripetal acceleration at the lowest point does not exceed $$6.0\,g$$?

Accepted Answer

$$r = \dfrac{v^2$}{a_c}$$ where $$a_c = 6g$$

Question 10

Two blocks, with masses $$m_A$$ and $$m_B$$, are connected to each other and to a central post by cords. They rotate about the post at frequency $$f ($$revolutions per second) on a frictionless horizontal surface at distances $$r_A$$ and $$r_B$$ from the post. What is the tension in the segment of cord between the post and $$m_A$$?

Accepted Answer

$$T_A = (m_A + m_B) (2 \pi f)^2$ r_A$$

Question 11

A bucket of mass $$2.00 kg is $$whirled in a vertical circle of radius $$1.10 m. $$What is the minimum speed at the top of the circle so that the rope does not go slack?

Accepted Answer

$$v_{min} = \sqrt{g r}$$

Question 12

In a system where $$m_A is on a $$table connected by a string over a pulley to $$m_B = 2.0 kg$$, what value of $$m_A$$ will keep the system moving at constant speed if the coefficient of kinetic friction is $$0.30$$?

Accepted Answer

$$m_A = \dfrac{m_B}{\mu_k}$$

Question 13

Suppose the space shuttle is in orbit $$400 km $$from Earth's surface, circling Earth every $$90$$ min. What is the centripetal acceleration in terms of $$g if$$ $$R_{Earth} = 6400 km$$?

Accepted Answer

$$a_c = g \left(\dfrac{R_{Earth}}{R_{Earth} + 400}\$$right)^2$$

Physics 170 Homework 4: Friction, Inclined Planes, Circular Motion, and Centripetal Force

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