Q: A horizontal beam is supported by a cable and a hinge at equilibrium. If the beam has mass $$m$$ and length $$L$$, and the cable makes an angle $$\theta$$ with the beam, what is the vertical force at the hinge?

$$F_{\text{hinge}} = mg - T \sin \theta$$

Q: A person holds a weight with their forearm parallel to the ground. If the weight is $$W$$, the distance from the elbow to the weight is $$d$$, and the muscle attaches at distance $$d_m$$ from the elbow at angle $$\theta$$, what force $$F_m$$ must the muscle exert?

$$F_m = \frac{W d}{d_m \sin \theta}$$

Question 1

A solid disk of mass $$m$$ and radius $$r is $$rotating about its center. If a net torque $$	au is $$applied, what is the angular acceleration $$\alpha of $$the disk?

Accepted Answer

$$\alpha = \frac{	au}{I}$$

Question 2

A windmill with blades of length $$L is $$rotating at angular velocity $$\omega. If $$the tip of a blade breaks off at the topmost point, and the center of the windmill is at height $$h$$ above the ground, where will the tip land horizontally from the base of the windmill? (Neglect air resistance.)

Accepted Answer

$$x = L \omega \sqrt{\frac{2h}{g}}$$

Question 3

A dinner plate of mass $$M is $$dropped and breaks into three pieces. Piece 1 (mass $$m_1) $$moves east at velocity $$v_1$$, piece 2 (mass $$m_2) $$moves north at velocity $$v_2. $$What is the direction of velocity of piece 3 (mass $$m_3$$)?

Accepted Answer

Southwest

Question 4

A horizontal beam is supported by a cable and a hinge at equilibrium. If the beam has mass $$m$$ and length $$L$$, and the cable makes an angle $$	heta$$ with the beam, what is the vertical force at the hinge?

Accepted Answer

$$F_{	ext{hinge}} = mg - T \sin 	heta$$

Question 5

A car traveling at $$v_0 is $$being followed by another car at distance $$d. $$The second car accelerates at $$a to $$catch up. How long does it take for the second car to reach the first?

Accepted Answer

$$t = \frac{-v_0$ + \sqrt{v_0$^2 + 2ad}}{a}$$

Question 6

A container full of liquid has a faucet at the bottom. If the height of the liquid above the faucet is $$h$$, what is the speed $$v of $$the liquid exiting the faucet? (Ignore spout diameter.)

Accepted Answer

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

Question 7

What is the average kinetic energy $$K_{avg} of a $$molecule in an ideal gas at temperature $$T$$?

Accepted Answer

$$K_{avg} = \frac{3}{2}k_B T$$

Question 8

A pendulum bob is released from height $$h. $$What is its speed $$v at $$the lowest point?

Accepted Answer

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

Question 9

Given the root mean squared speed $$v_{rms} of a $$gas molecule and its molar mass $$M$$, what is the temperature $$T of $$the gas?

Accepted Answer

$$T = \frac{M v_{rms}^2}{3R}$$

Question 10

How much heat $$Q is $$required to raise a block of ice of mass $$m$$ from temperature $$T_1 ($$below $$0^\circC) to $$water at temperature $$T_2 ($$above $$0^\circC$$)? Given $$c_{ice}$$, $$c_{water}$$, and $$L_f.$$

Accepted Answer

$$Q = m c_{ice} (0 - T_1$) + m L_f + m c_{water} (T_2$ - 0)$$

Question 11

An object is held by three wires: two wires pull up at angles $$\$$theta_1$$ and $$\$$theta_2$$, and one wire pulls down. If the object's weight is $$W$$, what is the tension $$T_1 in $$the first wire?

Accepted Answer

$$T_1$$ = \frac{W \sin \$$theta_2$$}{\sin \$$theta_1$$ + \sin \$$theta_2$$}$$

Question 12

How much heat is needed to convert ice at $$-T to $$water at $$0^\circC$$?

Accepted Answer

$$Q = m c_{ice} T + m L_f$$

Question 13

A person holds a weight with their forearm parallel to the ground. If the weight is $$W$$, the distance from the elbow to the weight is $$d$$, and the muscle attaches at distance $$d_m$$ from the elbow at angle $$	heta$$, what force $$F_m$$ must the muscle exert?

Accepted Answer

$$F_m = \frac{W d}{d_m \sin 	heta}$$

Question 14

A refrigerator removes $$Q_{in}$$ heat from inside and expels $$Q_{out}$$ heat outside. What is the work $$W$$ done by the refrigerator?

Accepted Answer

$$W = Q_{out} - Q_{in}$$

Question 15

A cart moves in a vertical loop of radius $$R. $$What is the minimum speed $$v_{min} at $$the top of the loop to keep the cart on the track?

Accepted Answer

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

Question 16

A Ferris wheel slows down from angular velocity $$\$$omega_0$$ with angular acceleration $$\alpha$$ (negative). How many revolutions N$ does it make before stopping?

Accepted Answer

$$N = \frac{\$$omega_0$$^2}{4 \pi |\alpha|}$$

Question 17

Two blocks collide on a frictionless surface. Block 1 (mass $$m_1) $$moves at $$v_1$$ and block 2 (mass $$m_2) is $$stationary. If the collision lasts time $$\Delta t$$ and block 1 rebounds with velocity $$v_1$$'$$, what is the average force F$ during the collision?

Accepted Answer

$$F = \frac{m_1$ (v_1$ - v_1$')}{\Delta t}$$

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