Q: In the system shown in image 2, block $$m_1$$ hangs vertically while block $$m_2$$ rests on a frictionless incline at angle $$\theta. If$$ $$m_1$$ \u003e $$m_2$$, which of the following best describes the acceleration of the system?

$$a = \frac{m_1$ g - m_2$ g \sin\theta}{m_1$ + m_2$}$$

Q: An airplane is flying in a horizontal circle at a speed $$v$$ with its wings tilted at an angle $$\theta to $$the horizontal (see image 4). If the lift force is perpendicular to the wings, what is the radius $$r of $$the circle?

$$r = \frac{v^2$}{g \tan\theta}$$

Question 1

A block of mass $$m$$ slides with velocity $$v on a $$frictionless surface and collides inelastically with a stationary block of mass $$3m in $$front of a spring (see image 1). After the collision, the combined mass compresses the spring until it comes to rest. If the spring constant is $$k$$, what is the maximum compression $$x_{max} of $$the spring?

Accepted Answer

$$x_{max} = \sqrt{\frac{2m v^2$}{4k}}$$

Question 2

In the system shown in image 2, block $$m_1$$ hangs vertically while block $$m_2$$ rests on a frictionless incline at angle $$	heta. If$$ $$m_1$$ \u003e $$m_2$$, which of the following best describes the acceleration of the system?

Accepted Answer

$$a = \frac{m_1$ g - m_2$ g \sin	heta}{m_1$ + m_2$}$$

Question 3

A roller coaster car of mass $$m$$ starts from rest at height $$h$$ and travels through a frictionless loop of radius $$R ($$see image 3). What is the speed of the car at point $$C$$, which is halfway up the loop (height $$h_C = R$$)?

Accepted Answer

$$v_C = \sqrt{2g(h - R)}$$

Question 4

An airplane is flying in a horizontal circle at a speed $$v$$ with its wings tilted at an angle $$	heta to $$the horizontal (see image 4). If the lift force is perpendicular to the wings, what is the radius $$r of $$the circle?

Accepted Answer

$$r = \frac{v^2$}{g 	an	heta}$$

Question 5

A uniform disk of mass $$3.5 kg is $$acted on by three forces as shown in image 5. What is the net torque about the center due to these forces?

Accepted Answer

$$	au = -17.9 N$$·m

Question 6

A uniform log of mass $$M is $$suspended by two wires, A and B, as shown in image 6. If $$d_A = 40 cm $$and $$d_B = 29 cm$$, what is the tension in wire B?

Accepted Answer

$$T_B = \frac{d_A}{d_A + d_B} Mg$$

Question 7

A rigid structure consists of a circular hoop of mass $$m$$ and radius $$2R$$ and a square made of four thin bars, each of length $$R$$ and mass $$m ($$see image 7). What is the total moment of inertia about the vertical axis?

Accepted Answer

$$I = m(4R^2$) + 4 \left(\frac{1}{12}mR^2$ + m\left(\frac{R}{2}\$$right)^2$$\right)$$

Question 8

An elephant of mass $$2000 kg $$stands on a piston of area $$2500 cm$^2 in a $$hydraulic lift (see image 8). If the small piston has area $$15 cm$^2$$, what force must be applied to the small piston to lift the elephant (ignore height difference)?

Accepted Answer

$$F = \frac{A_{small}}{A_{large}} mg$$

Question 9

In the system shown in image 9, a spherical shell of mass $$M_1$$ and radius $$R_1$$ rotates about a vertical axis. A massless cord passes over a pulley of mass $$m_2$$ and radius $$r_2$$, attached to a mass $$m_3. If$$ $$m_3$$ falls a distance $$h$$, what is its speed $$v$$ after falling?

Accepted Answer

$$v = \sqrt{\frac{2m_3gh}{m_3$ + \frac{I_1$}{R_1$^2} + \frac{I_2$}{r_2$^2}}}$$

Question 10

A 20 g block oscillates in simple harmonic motion on a spring, as shown in image 10. If the period is $$T = 80 ms$$, what is the maximum speed of the block if the amplitude is $$8 cm$$?

Accepted Answer

$$v_{max} = \omega A = \frac{2\pi}{T}A$$

Question 11

A 35 g block oscillates in simple harmonic motion on a spring, as shown in image 11. If the period is $$T = 50 ms$$, what is the spring constant $$k$$?

Accepted Answer

$$k = \frac{4\pi^2$ m}{T^2$}$$

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