Q: A box of mass $$m_1$$ hangs from the end of a uniform beam of length $$L$$ and mass $$m_2$$, which is supported at an angle $$\theta by a $$hinge and a horizontal rope attached a distance $$d$$ from the hinge. What is the tension $$T in $$the rope?

$$T = \frac{m_1$ g L \sin(\theta) + m_2$ g \frac{L}{2} \sin(\theta)}{d \cos(\theta)}$$

Question 1

A small ball of mass $$3m is $$moving with speed $$2v_0$$ toward a uniform disk of mass $$2m$$ and radius $$R ($$moment of inertia $$I = \frac{1}{2}MR^2$ with M = 2m$), which is moving to the left with velocity v_0$. After the ball sticks to the edge of the disk, what is the y$-coordinate of the center of mass of the combined system, taking y = 0$ at the bottom edge of the disk?

Accepted Answer

$$y_{cm} = \frac{2m \cdot 0 + 3m \cdot R}{5m}$$

Question 2

After the collision described above, what is the moment of inertia of the system (disk plus ball) about the system's center of mass?

Accepted Answer

$$I_{total} = \frac{1}{2} \cdot 2m \cdot R^2$ + 3m \cdot (R - y_{cm})^2$ + 2m \cdot (y_{cm})^2$

Question 3

What is the velocity of the center of mass of the disk-ball system after the collision, in terms of $$m$$, $$v_0$$, and direction?

Accepted Answer

$$v_{cm} = \frac{2m \cdot (-v_0$) + 3m \cdot 2v_0}{5m}$$, to the right

Question 4

A box of mass $$m_1$$ hangs from the end of a uniform beam of length $$L$$ and mass $$m_2$$, which is supported at an angle $$	heta by a $$hinge and a horizontal rope attached a distance $$d$$ from the hinge. What is the tension $$T in $$the rope?

Accepted Answer

$$T = \frac{m_1$ g L \sin(	heta) + m_2$ g \frac{L}{2} \sin(	heta)}{d \cos(	heta)}$$

Question 5

For the beam system above, what are the horizontal and vertical components of the force exerted by the hinge on the beam?

Accepted Answer

$$F_x = -T$$, $$F_y = m_1$ g + m_2$ g$$

Question 6

An ideal gas with adiabatic index $$\gamma = 2 is $$taken through a cycle as shown in the PV diagram. For the isobaric compression from state 1 to 2, what is the work done on the system?

Accepted Answer

$$W = P_0$ (V_0$ - V_2$)$$

Question 7

For the same gas cycle, what is the efficiency $$\eta of $$the heat engine in terms of pure numbers?

Accepted Answer

$$\eta = \frac{W_{net}}{Q_{in}}$$

Question 8

A heat engine operates between a hot reservoir at $$T_H = 500\,	ext{K}$$ and a cold reservoir at $$T_C = 200\,	ext{K}. $$What is the maximum possible efficiency for a Carnot engine operating between these temperatures?

Accepted Answer

$$\eta_{Carnot} = 1 - \frac{T_C}{T_H}$$

Question 9

A 0.5 kg block of ice at $$-12^\circC is $$placed in an insulated box containing 0.10 kg of steam at $$100^\circC. $$What is the total amount of energy required to melt the block of ice?

Accepted Answer

$$Q = m_{ice} c_{ice} \Delta T + m_{ice} L_f$$

Question 10

In the same calorimetry problem, what quantity of steam will have condensed into water at $$100^\circC $$due to the extraction of energy by heat that went into melting the ice?

Accepted Answer

$$m_{steam} = \frac{Q_{melt}}{L_v}$$

Question 11

On a frictionless frozen lake, a point mass $$2m$$ traveling at velocity $$v_0$$ collides elastically with one end of a stationary dumbbell consisting of two point masses $$m$$ attached by a massless rod of length $$L. Is $$both linear and angular momentum conserved in the collision?

Accepted Answer

Both linear and angular momentum are conserved.

Question 12

For the dumbbell system above, what is the moment of inertia of the dumbbell about its center of mass?

Accepted Answer

$$I = \frac{1}{2}mL^2$

Question 13

A worker of mass $$m$$ climbs a ladder of length $$L$$ and mass $$M_t$$, leaning against a frictionless wall at angle $$	heta. $$The coefficient of static friction between the ladder and the ground is $$\mu. $$What is the maximum distance $$x$$ the worker can climb before the ladder starts to slide?

Accepted Answer

$$x = \frac{\mu (m + M_t) L \cos 	heta}{m \sin 	heta + M_t \frac{L}{2} \sin 	heta}$$

Question 14

If the worker places a weight $$W_{add} at $$the bottom of the ladder to prevent slipping, what is the minimum weight required so the worker can climb to the top?

Accepted Answer

$$W_{add} = \frac{M_t}{2}g \frac{\cos 	heta}{\mu} + m g \frac{\cos 	heta}{\mu} - m g - M_t g$$

Question 15

A 120-g iron horseshoe at $$250^\circC is $$added to a 200-g copper calorimeter containing 0.5 kg of water at $$25^\circC. $$What is the final equilibrium temperature?

Accepted Answer

$$T_f = \frac{m_{Fe} c_{Fe} T_{Fe} + m_{Cu} c_{Cu} T_{Cu} + m_{w} c_{w} T_{w}}{m_{Fe} c_{Fe} + m_{Cu} c_{Cu} + m_{w} c_{w}}$$

Question 16

After equilibrium is established, ice at $$-5^\circC is $$added to the calorimeter. What is the minimum mass of ice needed for the final temperature to be exactly $$0^\circC$$?

Accepted Answer

$$m_{ice} = \frac{m_{Cu} c_{Cu} (T_f - 0) + m_{w} c_{w} (T_f - 0)}{L_f + c_{ice} (0 - T_{ice})}$$

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