Question 1

A solid sphere of mass $$M$$ and radius $$R$$ rolls down a ramp of height $$h$$ without slipping. What is the speed $$v of $$the sphere as it leaves the ramp?

Accepted Answer

$$v = \sqrt{\frac{10gh}{7}}$$

Question 2

For the rolling sphere above, what is the angular speed $$\omega_{s} of $$the sphere at the top of its trajectory in the air?

Accepted Answer

$$\omega_{s} = \frac{v}{R}$$

Question 3

When the sphere collides with a block at rest, which conservation law applies if the collision is perfectly elastic?

Accepted Answer

Conservation of momentum and kinetic energy

Question 4

A potter's wheel (solid disk, mass $$M$$, radius $$R) $$rotates at angular speed $$\$$omega_0$$. If a lump of clay (mass m$) sticks to the rim, what is the new angular speed $$\omega_f$$?

Accepted Answer

$$\omega_f = \$$omega_0$$ \frac{MR^2$}{MR^2$ + mR^2$}$$

Question 5

Which physical quantity is conserved when the clay sticks to the wheel (assuming no external torques)?

Accepted Answer

Angular momentum

Question 6

A beam is held in equilibrium by a sampling rope and a hinge. If the beam makes an angle $$	heta$$ with the horizontal, which force diagram correctly represents the forces acting on the beam?

Accepted Answer

Gravity, tension in the sampling rope, and hinge force at the pivot

Question 7

Given a fluid flowing from a large tank through a pipe, what is the approximate speed $$v_A of $$the fluid at the surface of the tank?

Accepted Answer

$$v_A \approx 0$$

Question 8

Using Bernoulli's equation, what is the speed $$v_D of $$the fluid at point D in the pipe if the height difference between the tank and D is $$h$$?

Accepted Answer

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

Question 9

A thin ring of mass $$M$$ and radius $$R is $$suspended and a magnet of mass $$m$$ collides with it tangentially. What is the rotational inertia $$I of $$the ring-magnet system about the suspension point?

Accepted Answer

$$I = MR^2$ + mR^2$

Question 10

After the collision, what is the angular velocity $$\omega of $$the ring-magnet system if the magnet sticks to the ring?

Accepted Answer

$$\omega = \frac{mv_0R}{MR^2$ + mR^2$}$$

Question 11

For small angle oscillations, what is the angular frequency $$\omega of $$the ring-magnet system?

Accepted Answer

$$\omega = \sqrt{\frac{MgR}{I}}$$

Question 12

A tightrope of length $$L$$ and tension $$T$$ supports a standing wave with wavelength $$\lambda. $$What is the fundamental frequency $$f_1 of $$the rope?

Accepted Answer

$$f_1$$ = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$$

Question 13

If the tightrope is clamped at both ends and the standing wave has $$n$$ antinodes, what is the length $$L of $$the rope in terms of the wavelength $$\lambda$$?

Accepted Answer

$$L = n \frac{\lambda}{2}$$

Question 14

A wave on a rope is described by $$y(x,t) = A \sin(kx - \omega t + \phi). $$What is the acceleration $$a(x,t) of $$the rope in its up-down motion?

Accepted Answer

$$a(x,t) = -\$$omega^2$$ A \sin(kx - \omega t + \phi)$$

Question 15

If air resistance creates a damping torque $$	au_d = -b \frac{d	heta}{dt}$$, what is the time $$t_{1/2}$$ for the amplitude of oscillation to be halved?

Accepted Answer

$$t_{1/2} = \frac{I}{b} \ln 2$$

Question 16

In the fluid flow problem, what is the relationship between pressure and speed at two points along a streamline according to Bernoulli's equation?

Accepted Answer

$$P_1$$ + \frac{1}{2}\rho $$v_1^2$$ + \rho $$gh_1$$ = $$P_2$$ + \frac{1}{2}\rho $$v_2^2$$ + \rho $$gh_2$$

Question 17

A block is struck by a rolling sphere. If both objects are of equal mass and the collision is perfectly elastic, what is the speed of the block immediately after the collision?

Accepted Answer

Equal to the initial speed of the sphere

Question 18

In the equilibrium of the beam system, which condition must be satisfied for static equilibrium?

Accepted Answer

Sum of all forces and sum of all torques must be zero

Physics 7A Final Exam Study Guide: Mechanics, Rotational Motion, Fluids, Oscillations, and Waves

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