Q: A mass-spring system makes 30 complete oscillations in 6.0 seconds. What is the frequency of the oscillations?

$$f = \frac{30}{6.0} = 5.0\ \text{Hz}$$

Q: A mass-spring system has a period of 3.0 s. What is its frequency?

$$f = \frac{1}{3.0} = 0.33\ \text{Hz}$$

Q: A spring with a spring constant $$k = 50\ \text{N/m} is $$stretched by $$0.20\ \text{m}. $$What is the magnitude of the force exerted by the spring?

$$F = kx = 50 \times 0.20 = 10\ \text{N}$$

Q: A mass-spring system has an amplitude of $$0.30\ \text{m}$$ and a period of $$2.0\ \text{s}. $$How far does the mass travel in $$6.0\ \text{s}$$?

$$\text{Distance} = 3 \text{ periods} \times 4A = 3 \times 4 \times 0.30 = 3.6\ \text{m}$$

Q: A spring stretches $$0.08\ \text{m}$$ when a $$2.0\ \text{kg}$$ mass is suspended from it. What is the spring constant?

$$k = \frac{mg}{x} = \frac{2.0 \times 9.8}{0.08} = 245\ \text{N/m}$$

Q: A mass of $$0.5\ \text{kg} is $$attached to a spring with $$k = 100\ \text{N/m}$$ and stretched by $$0.10\ \text{m}. $$What is the maximum elastic potential energy?

$$E_{PE} = \frac{1}{2} k x^2$ = \frac{1}{2} \times 100 \times (0.10)^2$ = 0.5\ \text{J}$$

Q: A bullet of mass $$0.02\ \text{kg}$$ moving at $$18\ \text{m/s}$$ strikes a block of mass $$0.54\ \text{kg} at $$rest. After the collision, the bullet becomes embedded in the block. What is the speed of the bullet-block system after the collision?

$$v' = \frac{m_1$ v_1$}{m_1$ + m_2$} = \frac{0.02 \times 18}{0.02 + 0.54} = 0.65\ \text{m/s}$$

Q: A mass-spring system has a mass of $$0.25\ \text{kg}$$ and a spring constant of $$10\ \text{N/m}. $$What is the period of oscillation?

$$T = 2\pi \sqrt{\frac{0.25}{10}} = 0.99\ \text{s}$$

Question 1

A mass-spring system makes 30 complete oscillations in 6.0 seconds. What is the frequency of the oscillations?

Accepted Answer

$$f = \frac{30}{6.0} = 5.0\ 	ext{Hz}$$

Question 2

A mass-spring system has a period of 3.0 s. What is its frequency?

Accepted Answer

$$f = \frac{1}{3.0} = 0.33\ 	ext{Hz}$$

Question 3

A spring with a spring constant $$k = 50\ 	ext{N/m} is $$stretched by $$0.20\ 	ext{m}. $$What is the magnitude of the force exerted by the spring?

Accepted Answer

$$F = kx = 50 	imes 0.20 = 10\ 	ext{N}$$

Question 4

A mass-spring system has an amplitude of $$0.30\ 	ext{m}$$ and a period of $$2.0\ 	ext{s}. $$How far does the mass travel in $$6.0\ 	ext{s}$$?

Accepted Answer

$$	ext{Distance} = 3 	ext{ periods} 	imes 4A = 3 	imes 4 	imes 0.30 = 3.6\ 	ext{m}$$

Question 5

A spring stretches $$0.08\ 	ext{m}$$ when a $$2.0\ 	ext{kg}$$ mass is suspended from it. What is the spring constant?

Accepted Answer

$$k = \frac{mg}{x} = \frac{2.0 	imes 9.8}{0.08} = 245\ 	ext{N/m}$$

Question 6

At which location(s) is the magnitude of the velocity in a mass-spring system a maximum?

Accepted Answer

$$x = 0$$

Question 7

A mass of $$0.5\ 	ext{kg} is $$attached to a spring with $$k = 100\ 	ext{N/m}$$ and stretched by $$0.10\ 	ext{m}. $$What is the maximum elastic potential energy?

Accepted Answer

$$E_{PE} = \frac{1}{2} k x^2$ = \frac{1}{2} 	imes 100 	imes (0.10)^2$ = 0.5\ 	ext{J}$$

Question 8

A mass-spring system has a mass of $$1.0\ 	ext{kg}$$ and a spring constant of $$25\ 	ext{N/m}. $$What is the period of oscillation?

Accepted Answer

$$T = 2\pi \sqrt{\frac{m}{k}} = 2\pi \sqrt{\frac{1.0}{25}} = 1.26\ 	ext{s}$$

Question 9

A simple pendulum with a length of $$1.5\ 	ext{m}$$ oscillates near the surface of the Earth. What is its period?

Accepted Answer

$$T = 2\pi \sqrt{\frac{L}{g}} = 2\pi \sqrt{\frac{1.5}{9.8}} = 2.46\ 	ext{s}$$

Question 10

A bullet of mass $$0.02\ 	ext{kg}$$ moving at $$18\ 	ext{m/s}$$ strikes a block of mass $$0.54\ 	ext{kg} at $$rest. After the collision, the bullet becomes embedded in the block. What is the speed of the bullet-block system after the collision?

Accepted Answer

$$v' = \frac{m_1$ v_1$}{m_1$ + m_2$} = \frac{0.02 	imes 18}{0.02 + 0.54} = 0.65\ 	ext{m/s}$$

Question 11

A mass-spring system has a mass of $$0.25\ 	ext{kg}$$ and a spring constant of $$10\ 	ext{N/m}. $$What is the period of oscillation?

Accepted Answer

$$T = 2\pi \sqrt{\frac{0.25}{10}} = 0.99\ 	ext{s}$$

Question 12

A simple pendulum with a length of $$2.0\ 	ext{m}$$ oscillates near the surface of the Earth. What is its frequency?

Accepted Answer

$$f = \frac{1}{T} = \frac{1}{2\pi \sqrt{\frac{2.0}{9.8}}} = 0.36\ 	ext{Hz}$$

Question 13

Which of the following factors affect the period of a simple pendulum?

Accepted Answer

I and II (acceleration due to gravity and length of the string)

Question 14

A mass-spring system has a mass of $$2.0\ 	ext{kg}$$, a spring constant of $$200\ 	ext{N/m}$$, and an amplitude of $$3.0\ 	ext{m}. $$What is the total energy of the system?

Accepted Answer

$$E_{total} = \frac{1}{2} k A^2$ = \frac{1}{2} 	imes 200 	imes (3.0)^2$ = 900\ 	ext{J}$$

Question 15

A mass-spring system has a mass of $$0.15\ 	ext{kg}$$ and a spring constant of $$30.2\ 	ext{N/m}. $$What is the period of oscillation?

Accepted Answer

$$T = 2\pi \sqrt{\frac{0.15}{30.2}} = 0.44\ 	ext{s}$$

Question 16

A mass of $$0.062\ 	ext{kg}$$ oscillates on a simple pendulum with a length of $$1.6\ 	ext{m}$$ and reaches a maximum height of $$0.08\ 	ext{m}. $$What is the maximum speed of the mass?

Accepted Answer

$$v_{max} = \sqrt{2gh_{max}} = \sqrt{2 	imes 9.8 	imes 0.08} = 1.25\ 	ext{m/s}$$

Question 17

A bullet-block system (total mass $$0.56\ 	ext{kg}) $$attached to a spring ($$k = 80\ 	ext{N/m}) $$comes to rest after a collision. If the maximum elastic potential energy is $$0.116\ 	ext{J}$$, what is the maximum compression of the spring?

Accepted Answer

$$x = \sqrt{\frac{2E_{PE}}{k}} = \sqrt{\frac{2 	imes 0.116}{80}} = 0.054\ 	ext{m}$$

Simple Harmonic Motion: Mass-Spring Systems and Pendulums

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