Q: A box of mass $$m_1$$ hangs from the end of a uniform beam of length $$L$$ and mass $$m_2$$, 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?

$$T = \frac{m_1gL\sin\theta + m_2g\frac{L}{2}\sin\theta}{d\cos\theta}$$

Q: For the beam system above, if the rope is cut, what is the initial angular acceleration $$\alpha of $$the beam about the hinge?

$$\alpha = \frac{m_1gL\sin\theta + m_2g\frac{L}{2}\sin\theta}{\frac{1}{3}m_2L^2$}$$

Question 1

A small ball of mass $$3m$$ moves at 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$), which is moving to the left with velocity v_0$. The ball grazes the edge of the disk and sticks to it. What is the y$-coordinate of the center of mass of the system after the collision, with y=0$ at the bottom edge of the disk?

Accepted Answer

$$y_{cm} = \frac{3R}{5}$$

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}(2m)R^2$ + 3m(R - y_{cm})^2$

Question 3

What is the velocity of the center of mass of the disk-ball system after the collision? (Indicate both magnitude and direction.)

Accepted Answer

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

Question 4

A box of mass $$m_1$$ hangs from the end of a uniform beam of length $$L$$ and mass $$m_2$$, 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

$$T = \frac{m_1gL\sin	heta + m_2g\frac{L}{2}\sin	heta}{d\cos	heta}$$

Question 5

For the beam system above, if the rope is cut, what is the initial angular acceleration $$\alpha of $$the beam about the hinge?

Accepted Answer

$$\alpha = \frac{m_1gL\sin	heta + m_2g\frac{L}{2}\sin	heta}{\frac{1}{3}m_2L^2$}$$

Question 6

An ideal gas with adiabatic index $$\gamma = 2$$ undergoes an isobaric compression from $$V_0 to$$ $$V_0/2$ at pressure P_0$. What is the work done on the gas during this process?

Accepted Answer

$$W = P_0$(V_0$/2 - V_0$) = -\frac{P_0V_0}{2}$$

Question 7

In the thermodynamic cycle shown, what is the efficiency $$\eta of $$the heat engine in terms of the temperatures $$T_H$$ and $$T_C$$?

Accepted Answer

$$\eta = 1 - \frac{T_C}{T_H}$$

Question 8

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

Accepted Answer

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

Question 9

A spacecraft orbits Mars in an ellipse with periareon distance $$r_p$$ and apoareon distance $$r_a. $$What is the ratio of its speed at periareon $$v_p to $$its speed at apoareon $$v_a$$?

Accepted Answer

$$\frac{v_p}{v_a} = \frac{r_a}{r_p}$$

Question 10

A uniform rod of mass $$M$$ and length $$L is $$struck at one end by a point mass $$2m$$ moving at velocity $$v_0 on a $$frictionless surface. The dumbbell consists of two masses $$m$$ separated by $$L. $$What is the moment of inertia of the dumbbell about its center of mass?

Accepted Answer

$$I = 2m\left(\frac{L}{2}\$$right)^2$$ = \frac{1}{2}mL^2$

Question 11

In a calorimetry experiment, a 120-g iron horseshoe at $$250^\circC is $$added to 0.5 kg of water at $$25^\circC in a 200-g $$copper calorimeter. What is the equation for the final equilibrium temperature $$T_f$$?

Accepted Answer

$$(m$$_{Fe}$$c_{Fe})(T_f - T_{Fe,i}) + (m$$_{Cu}$$c_{Cu})(T_f - T_{Cu,i}) + (m$$_{H_2O}$$c_{H_2O})(T_f - T_{H_2O,i}) = 0$$

Question 12

A heat engine absorbs $$Q_H$$ from a hot reservoir at $$T_H$$ and expels $$Q_C to a $$cold reservoir at $$T_C. $$Which of the following must be true for the engine to be possible according to the second law of thermodynamics?

Accepted Answer

$$\frac{Q_C}{Q_H} \geq \frac{T_C}{T_H}$$

Question 13

A worker of mass $$m$$ climbs a ladder of length $$L$$ and mass $$M$$ leaning against a frictionless wall at angle $$	heta. $$The coefficient of static friction at the base is $$\mu. $$What is the maximum distance $$x$$ the worker can climb before the ladder slips?

Accepted Answer

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

Question 14

In the thermodynamic cycle shown, which process corresponds to an isochoric heating?

Accepted Answer

2–3

Question 15

A ball of mass $$M$$ and radius $$R is $$spinning with angular velocity $$\omega$$ and moving at velocity $$v_0$$ before colliding and sticking to a rod of mass $$M$$ and length $$L. If $$after the collision the system does not rotate, what was $$\omega$$ before the collision?

Accepted Answer

$$\omega = \frac{v_0$ L}{2R}$$

Question 16

A heat engine operates between $$T_H = 500\ \mathrm{K}$$ and $$T_C = 200\ \mathrm{K}. $$What is the maximum possible efficiency for any engine operating between these temperatures?

Accepted Answer

$$\eta_{max} = 1 - \frac{T_C}{T_H} = 0.6$$

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