Q: Which equation correctly expresses angular momentum as a cross product for a particle of mass $$m$$ moving with velocity $$\vec{v} at $$position $$\vec{r}$$?

$$\vec{L} = \vec{r} \times m \vec{v}$$

Q: For a simple pendulum of length $$L$$ and mass $$m$$, what is the angular acceleration $$\alpha at an $$angle $$\theta$$ from the vertical?

$$\alpha = -\frac{g}{L} \sin \theta$$

Q: Using the small-angle approximation for a pendulum, which equation best describes the angular acceleration?

$$\alpha = -\frac{g}{L} \theta$$

Question 1

An ice skater spins with arms extended and then pulls her arms in close to her body. Which of the following statements best explains what happens to her angular velocity, assuming no external torque acts on her?

Accepted Answer

Her angular velocity increases because her moment of inertia decreases.

Question 2

A powered toy car is initially at rest on a track with bearings on the top and bottom of the axis, so there is no external torque in the horizontal plane. When the car starts to accelerate, what happens to the angular velocity of the track?

Accepted Answer

It rotates in the opposite direction to the car.

Question 3

Which equation correctly expresses angular momentum as a cross product for a particle of mass $$m$$ moving with velocity $$\vec{v} at $$position $$\vec{r}$$?

Accepted Answer

$$\vec{L} = \vec{r} 	imes m \vec{v}$$

Question 4

Two ice skaters of equal mass $$m$$ collide inelastically and grab each other's hands, rotating about their combined center of mass. If their initial velocities are $$\vec{v}$$ and $$-\vec{v}$$, what is the magnitude of their total angular momentum after the collision?

Accepted Answer

$$L_T = 2mr^2$ \omega_{f}$$

Question 5

A mass $$m is $$attached to a spring with spring constant $$k$$ and oscillates horizontally with negligible friction. What is the period $$T of $$the oscillation?

Accepted Answer

$$T = 2\pi \sqrt{\frac{m}{k}}$$

Question 6

You want to measure the mass $$m of an $$object using a spring with known spring constant $$k by $$timing its oscillations. Which formula gives $$m in $$terms of the measured period $$T$$?

Accepted Answer

$$m = \frac{kT^2$}{4\pi^2$}$$

Question 7

A vertical spring supports a mass $$m. $$What is the equilibrium displacement $$y_0$$ from the unstretched position due to gravity?

Accepted Answer

$$y_0$$ = \frac{mg}{k}$$

Question 8

For a simple pendulum of length $$L$$ and mass $$m$$, what is the angular acceleration $$\alpha at an $$angle $$	heta$$ from the vertical?

Accepted Answer

$$\alpha = -\frac{g}{L} \sin 	heta$$

Question 9

Using the small-angle approximation for a pendulum, which equation best describes the angular acceleration?

Accepted Answer

$$\alpha = -\frac{g}{L} 	heta$$

Question 10

A physical pendulum consists of a rigid body of mass $$m$$ pivoted at a distance $$r$$ from its center of mass. What is the period $$T of $$small oscillations?

Accepted Answer

$$T = 2\pi \sqrt{\frac{I}{mgr}}$$

Question 11

Which of the following is the general solution to the wave equation $$\frac{d^2$ y}{dt^2$} = v^2$ \frac{d^2$ y}{dx^2$}$$ for a wave traveling in the $$x$$ direction?

Accepted Answer

$$y(x,t) = A \sin(kx - \omega t + \phi)$$

Question 12

For a wave on a string with linear density $$\mu$$ and tension $$T_{max}$$, what is the velocity $$v of $$the wave?

Accepted Answer

$$v = \sqrt{\frac{T_{max}}{\mu}}$$

Question 13

If you double the mass $$m$$ hanging from a wire and study standing waves, what happens to the frequency and wavelength of the fundamental mode?

Accepted Answer

Frequency doubles, wavelength unchanged.

Question 14

In a tube closed at one end and open at the other, what is the wavelength $$\lambda of $$the fundamental harmonic?

Accepted Answer

$$\lambda = 4L$$

Question 15

A sound wave travels through air. What is true about the displacement of air particles and the pressure variation?

Accepted Answer

The displacement maximum is a quarter period out of phase with the pressure maximum.

Question 16

A stationary observer hears a higher pitch from an approaching ambulance. Which effect explains this observation?

Accepted Answer

The Doppler effect, with the source moving toward the observer.

Question 17

Which equation gives the frequency $$f$$ heard by a stationary observer when a sound source moves toward them at speed $$v_s in a $$medium where the speed of sound is $$v$$?

Accepted Answer

$$f = f_0$ \frac{v}{v - v_s}$$

Question 18

Which formula correctly expresses the relativistic kinetic energy $$K of a $$particle of mass $$m$$ moving at speed $$v$$?

Accepted Answer

$$K = (\gamma - 1)mc^2$

Question 19

A muon created in the upper atmosphere travels toward Earth at nearly the speed of light. If its proper lifetime is $$2.2 \mu s$$ and the Lorentz factor $$\gamma is 100$$, what is its observed lifetime in Earth's frame?

Accepted Answer

$$220 \mu s$$

Physics 211: Angular Momentum, Oscillations, Waves, Sound, and Special Relativity

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