Lesson test: Small oscillations

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Small oscillations

1.[3p]
Why does every stable equilibrium behave like a spring for small displacements?

The Taylor expansion of the potential has a vanishing first derivative at a minimum, leaving a quadratic term

Correct

Every restoring force in nature is exactly linear

The answer is: The Taylor expansion of the potential has a vanishing first derivative at a minimum, leaving a quadratic term

All materials obey Hooke's law

The answer is: The Taylor expansion of the potential has a vanishing first derivative at a minimum, leaving a quadratic term

The potential energy is always zero at equilibrium

The answer is: The Taylor expansion of the potential has a vanishing first derivative at a minimum, leaving a quadratic term

2.[2p]
A 0.80 kg mass on a spring of stiffness 320 N m⁻¹ oscillates. What is the period, in seconds?

CorrectNot quite: 0.314

3.[2p]
A mass on a spring oscillates at 8.0 rad s⁻¹ with an amplitude of 0.050 m. What is its maximum speed, in metres per second?

CorrectNot quite: 0.40

4.[2p]
The period of a simple harmonic oscillator increases with the amplitude of the motion.

True

The answer is: False

False

Correct

5.[2p]
What is the period of a simple pendulum of length 0.60 m, in seconds, with $g = 9.81$ m s⁻²?

CorrectNot quite: 1.554

6.[3p]
A pendulum has a period of 2.00 s on Earth. What length is it, in metres?

CorrectNot quite: 0.994

7.[3p]
A uniform rod of length $L$ swings about one end. Which simple pendulum keeps the same time?

One of length $2 L / 3$

Correct

One of length $L$

The answer is: One of length $2L/3$

One of length $L / 2$

The answer is: One of length $2L/3$

One of length $3 L / 2$

The answer is: One of length $2L/3$

8.[3p]
Which of these are true of the small angle approximation for a pendulum?

Select all that apply

The exact period at an amplitude of 30° is about 1.7 per cent longer than the small angle value

Correct

The error grows roughly as the square of the amplitude

Correct

A clock escapement is valuable partly because it keeps the amplitude constant

Correct

The approximation makes the equation linear, which is what allows a closed form solution

Correct

9.[3p]
Match each damping condition to what the system does after a displacement.

No damping
Light damping
Critical damping
Heavy damping

creeps back to equilibrium slowly without oscillating
oscillates forever at constant amplitude
oscillates with an exponentially decaying amplitude
returns to equilibrium fastest without overshooting

Show the answer

No damping: oscillates forever at constant amplitude Light damping: oscillates with an exponentially decaying amplitude Critical damping: returns to equilibrium fastest without overshooting Heavy damping: creeps back to equilibrium slowly without oscillating

1.[3p] Why does every stable equilibrium behave like a spring for small displacements?

2.[2p] A 0.80 kg mass on a spring of stiffness 320 N m⁻¹ oscillates. What is the period, in seconds?

3.[2p] A mass on a spring oscillates at 8.0 rad s⁻¹ with an amplitude of 0.050 m. What is its maximum speed, in metres per second?

4.[2p] The period of a simple harmonic oscillator increases with the amplitude of the motion.

5.[2p] What is the period of a simple pendulum of length 0.60 m, in seconds, with g=9.81 m s⁻²?

6.[3p] A pendulum has a period of 2.00 s on Earth. What length is it, in metres?

7.[3p] A uniform rod of length L swings about one end. Which simple pendulum keeps the same time?

8.[3p] Which of these are true of the small angle approximation for a pendulum?

9.[3p] Match each damping condition to what the system does after a displacement.

1.[3p]
Why does every stable equilibrium behave like a spring for small displacements?

2.[2p]
A 0.80 kg mass on a spring of stiffness 320 N m⁻¹ oscillates. What is the period, in seconds?

3.[2p]
A mass on a spring oscillates at 8.0 rad s⁻¹ with an amplitude of 0.050 m. What is its maximum speed, in metres per second?

4.[2p]
The period of a simple harmonic oscillator increases with the amplitude of the motion.

5.[2p]
What is the period of a simple pendulum of length 0.60 m, in seconds, with $g = 9.81$ m s⁻²?

6.[3p]
A pendulum has a period of 2.00 s on Earth. What length is it, in metres?

7.[3p]
A uniform rod of length $L$ swings about one end. Which simple pendulum keeps the same time?

8.[3p]
Which of these are true of the small angle approximation for a pendulum?

9.[3p]
Match each damping condition to what the system does after a displacement.