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 same sphere, what is its angular speed $$\omega_{s} at $$the top of its trajectory after leaving the ramp?

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 both momentum and kinetic energy

Question 4

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

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 rope making an angle $$	heta$$ with the beam. What is the condition for equilibrium in terms of forces and torques?

Accepted Answer

Both the sum of all forces and the sum of all torques must be zero

Question 7

In a fluid system with two large tanks connected by a pipe, what principle is used to relate the speed and pressure at different points in the pipe?

Accepted Answer

Bernoulli's equation

Question 8

Given a vertical tube connected to a horizontal pipe, how do you determine the height $$h of $$the fluid in the tube?

Accepted Answer

$$h = \frac{P}{\rho g}$$

Question 9

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

Accepted Answer

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

Question 10

After the magnet sticks to the ring, what is the angular velocity $$\omega of $$the system if the magnet was moving at speed $$v$$ just before collision?

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

$$f_1$$ = \frac{v}{2L}$$, where v$$ = \sqrt{\frac{T}{\mu}}$$

Question 14

What is the minimum amplitude $$A_{min} of $$the wave that would cause the bug to become momentarily 'weightless' at the crest?

Accepted Answer

$$A_{min} = \frac{g}{\$$omega^2$$}$$

Question 15

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

Accepted Answer

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

Question 16

In the fluid tank problem, if the cross-sectional area of the tank is very large, what is the approximate speed $$v_A of $$the fluid at point A?

Accepted Answer

$$v_A \approx 0$$

Question 17

If the tightrope is clamped at both ends and the wavelength $$\lambda is $$known, what is the length $$L of $$the rope for the $$nth $$harmonic?

Accepted Answer

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

Physics 7A Final Exam Study Guide: Rotational Dynamics, Fluid Mechanics, Oscillations, and Waves

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