Q: A block of mass $$m$$ slides down a rough incline of angle $$\theta$$ and length $$L. $$The coefficient of kinetic friction is $$\mu_k. $$What is the speed of the block at the bottom?

$$v = \sqrt{2gL(\sin\theta - \mu_k\cos\theta)}$$

Q: A block of mass $$m$$ slides down a rough incline and is connected to a pulley of mass $$M$$ and radius $$R. If $$the block descends a distance $$d$$, what is the work done by friction?

$$W_f = \mu_k m g d \sin\theta$$

Q: A block of mass $$m$$ slides down a rough incline and is connected to a pulley of mass $$M$$ and radius $$R. If $$the block descends a distance $$d$$, what is the change in mechanical energy due to friction?

$$\Delta E = -\mu_k m g d \sin\theta$$

Question 1

A person weighing 800 N climbs a uniform ladder that is 5.0 m long and weighs 200 N. The bottom of the ladder rests on the floor and the top against a frictionless wall. If the person stands 4.0 m from the base, what is the minimum coefficient of static friction required at the base to prevent slipping?

Accepted Answer

$$\mu_s = \frac{2}{5}$$

Question 2

A kid with mass 20 kg climbs up a plank of length 5.0 m and mass 12.0 kg, inclined at $$30^\circ to $$the horizontal. If the coefficient of static friction between the plank and the ground is $$\mu_s = 0.3$$, what is the maximum distance the kid can climb before the plank starts to slip?

Accepted Answer

$$d = 3.2\ \mathrm{m}$$

Question 3

A solid cylinder of mass $$M$$ and radius $$R is $$released from rest at the top of an incline of height $$h$$ and rolls down without slipping. What is its speed at the bottom?

Accepted Answer

$$v = \sqrt{\frac{2gh}{1 + \frac{1}{2}}}$$

Question 4

A primitive yo-yo consists of a massless string wrapped around a solid cylinder of mass $$M$$ and radius $$R. If $$the yo-yo is released from rest and descends a distance $$h$$, what is its final speed using energy conservation?

Accepted Answer

$$v = \sqrt{\frac{4gh}{3}}$$

Question 5

A block of mass $$m$$ slides down a rough incline of angle $$	heta$$ and length $$L. $$The coefficient of kinetic friction is $$\mu_k. $$What is the speed of the block at the bottom?

Accepted Answer

$$v = \sqrt{2gL(\sin	heta - \mu_k\cos	heta)}$$

Question 6

A solid sphere of mass $$m$$ and radius $$R$$ rolls without slipping down a ramp of height $$h. $$What fraction of its total mechanical energy is rotational kinetic energy at the bottom?

Accepted Answer

$$\frac{2}{5}$$

Question 7

A plank of mass $$M$$ and length $$L is $$pivoted at one end and held horizontally. If released from rest, what is the angular acceleration $$\alpha$$ just after release?

Accepted Answer

$$\alpha = \frac{3g}{2L}$$

Question 8

A block of mass $$m is $$attached to a pulley of mass $$M$$ and radius $$R$$ and released from rest. If the pulley is a solid disk, what is the acceleration of the block?

Accepted Answer

$$a = \frac{g}{1 + \frac{M}{2m}}$$

Question 9

A solid sphere of mass $$m$$ and radius $$R$$ starts from rest and rolls down a rough incline of height $$h$$ and then onto a smooth horizontal surface. What is the speed of the sphere at the bottom of the incline?

Accepted Answer

$$v = \sqrt{\frac{10gh}{7}}$$

Question 10

A block of mass $$m$$ slides down a rough incline and is connected to a pulley of mass $$M$$ and radius $$R. If $$the block descends a distance $$d$$, what is the work done by friction?

Accepted Answer

$$W_f = \mu_k m g d \sin	heta$$

Question 11

A uniform rod of length $$L$$ and mass $$M$$ rotates about an axis perpendicular to its length and passing through one end. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{3}ML^2$

Question 12

A solid cylinder of mass $$M$$ and radius $$R is $$rolling without slipping. What is the relationship between its translational velocity $$v$$ and angular velocity $$\omega$$?

Accepted Answer

$$v = \omega R$$

Question 13

A block of mass $$m$$ slides down a frictionless incline of angle $$	heta$$ and length $$L. $$What is the speed of the block at the bottom?

Accepted Answer

$$v = \sqrt{2gL\sin	heta}$$

Question 14

A solid sphere of mass $$m$$ and radius $$R$$ rolls without slipping down an incline. What is the ratio of its rotational kinetic energy to its total kinetic energy at the bottom?

Accepted Answer

$$\frac{2}{7}$$

Question 15

A plank of mass $$M$$ and length $$L is $$pivoted at one end and released from rest in a horizontal position. What is the speed of the free end as it reaches the vertical position?

Accepted Answer

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

Question 16

A block of mass $$m is $$attached to a string that passes over a pulley of mass $$M$$ and radius $$R. If $$the block descends a distance $$h$$, what is the final speed of the block?

Accepted Answer

$$v = \sqrt{\frac{2gh}{1 + \frac{M}{2m}}}$$

Question 17

A solid sphere of mass $$m$$ and radius $$R$$ rolls down a rough incline of height $$h. $$What is the total kinetic energy at the bottom?

Accepted Answer

$$K = mgh$$

Question 18

A block of mass $$m$$ slides down a rough incline and is connected to a pulley of mass $$M$$ and radius $$R. If $$the block descends a distance $$d$$, what is the change in mechanical energy due to friction?

Accepted Answer

$$\Delta E = -\mu_k m g d \sin	heta$$

Rotational Kinematics and Dynamics: Problem Solving and Applications

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