Q: A cylinder of mass $$M$$ and radius $$R$$ rests on a slope at angle $$\theta. $$The coefficient of static friction between the cylinder and the slope is $$\mu_s. $$What is the maximum angle $$\theta$$ before the cylinder begins to slip?

$$\theta_{max} = \arctan(\mu_s)$$

Question 1

Given three masses $$m_1$$, $$m_2$$, and $$m_3$$ arranged as shown in the diagram, with $$m_1$$ and $$m_3$$ separated by distance $$d$$, and $$m_2$$ located a distance $$d/2$$ above $$m_1$$, what are the coordinates of the center of mass of the system?

Accepted Answer

$$\left( \frac{m_1$ d + m_3$ d}{m_1$ + m_2$ + m_3$}, \frac{m_2$ d/2}{m_1$ + m_2$ + m_3$} \right)$$

Question 2

If the system of three masses rotates about the center of mass, which statement is true about the net torque acting on the system?

Accepted Answer

The net torque about the center of mass is zero.

Question 3

A system of three masses is rotating about a point that is not the center of mass. Which of the following is true about the kinetic energy of the system?

Accepted Answer

The kinetic energy is greater than if the system rotated about the center of mass.

Question 4

Which of the following best describes the moment of inertia of the system about the center of mass?

Accepted Answer

$$I = \sum m_i r_i^2$ where r_i$ is the distance from each mass to the center of mass.

Question 5

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 6

Two blocks, $$m_1$$ and $$m_2$$, are connected by a light string over a frictionless pulley. If $$m_1$$ \u003e $$m_2$$, what is the acceleration of the system?

Accepted Answer

$$a = \frac{(m_1$ - m_2$)g}{m_1$ + m_2$}$$

Question 7

A cylinder of mass $$M$$ and radius $$R$$ rests on a slope at angle $$	heta. $$The coefficient of static friction between the cylinder and the slope is $$\mu_s. $$What is the maximum angle $$	heta$$ before the cylinder begins to slip?

Accepted Answer

$$	heta_{max} = \arctan(\mu_s)$$

Question 8

A block of mass $$m$$ slides down a rough incline of height $$h$$ and length $$L$$ with coefficient of kinetic friction $$\mu_k. $$What is its speed at the bottom?

Accepted Answer

$$v = \sqrt{2g(h - \mu_k L)}$$

Question 9

If a system of masses is rotating about an axis, which of the following statements is true about the angular momentum?

Accepted Answer

Angular momentum is conserved if no external torque acts on the system.

Question 10

A block of mass $$m is $$released from rest at the top of a frictionless ramp of height $$h. If $$the block reaches the bottom with speed $$v$$, which equation correctly relates $$v to$$ $$h$$?

Accepted Answer

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

Question 11

A block of mass $$m is $$pushed up a frictionless incline with initial speed $$v_0. $$What is the maximum height $$h_{max} it $$reaches?

Accepted Answer

$$h_{max} = \frac{v_0$^2}{2g}$$

Question 12

A block of mass $$m$$ slides down a ramp and collides elastically with another block of mass $$M at $$rest. Which conservation law applies to the collision?

Accepted Answer

Both momentum and kinetic energy are conserved.

Question 13

A block of mass $$m is $$attached to a spring of constant $$k$$ and oscillates on a frictionless surface. What is the period of oscillation?

Accepted Answer

$$T = 2\pi \sqrt{\frac{m}{k}}$$

Question 14

A block of mass $$m$$ slides down a ramp of angle $$	heta$$ and length $$L. $$What is the acceleration of the block?

Accepted Answer

$$a = g \sin 	heta$$

Question 15

A cylinder of mass $$M$$ and radius $$R is $$attached to a rope and rests on a slope at angle $$	heta. If $$the coefficient of static friction is $$\mu_s$$, what is the condition for the cylinder to remain at rest?

Accepted Answer

$$	an 	heta \leq \mu_s$$

Question 16

A block of mass $$m is $$released from rest at the top of a ramp of height $$h$$ and length $$L. If $$the ramp is inclined at angle $$	heta$$, what is the time $$t it $$takes to reach the bottom?

Accepted Answer

$$t = \sqrt{\frac{2L}{g \sin 	heta}}$$

Question 17

A block of mass $$m$$ slides down a ramp and compresses a spring of constant $$k by $$distance $$x. $$What is the relationship between the initial potential energy and the spring energy at maximum compression?

Accepted Answer

$$mgh = \frac{1}{2}kx^2$

Physics 407: Rigid Body Motion, Center of Mass, and Static Equilibrium Study Notes

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