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 spring constant $$k = 50\ \text{N/m}$$ and is stretched by $$0.15\ \text{m}. $$What is the magnitude of the force exerted by the spring?

$$F = kx = 50 \times 0.15 = 7.5\ \text{N}$$

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

$$T = 2\pi \sqrt{\frac{1.5}{9.8}} \approx 2.46\ \text{s}$$

Q: A mass of $$0.25\ \text{kg} is $$attached to a spring with $$k = 10\ \text{N/m}. $$What is the period of oscillation?

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

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. What is the speed of the bullet-block system after the collision?

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

Q: A mass-spring system with mass $$m = 2.0\ \text{kg}$$ and spring constant $$k = 200\ \text{N/m}$$ has an amplitude $$A = 3.0\ \text{m}. $$What is the total energy of the system?

$$E = \frac{1}{2} k A^2$ = \frac{1}{2} \times 200 \times 3^2$ = 900\ \text{J}$$

Q: A mass-spring system with mass $$m = 0.5\ \text{kg}$$ and spring constant $$k = 120\ \text{N/m}$$ oscillates. What is its period?

$$T = 2\pi \sqrt{\frac{0.5}{120}} \approx 0.41\ \text{s}$$

Q: A simple pendulum of mass $$0.5\ \text{kg}$$ and length $$1.5\ \text{m}$$ reaches a maximum height of $$0.08\ \text{m}. $$What is the maximum gravitational potential energy?

$$U = mgh = 0.5 \times 9.8 \times 0.08 = 0.39\ \text{J}$$

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 spring constant $$k = 50\ 	ext{N/m}$$ and is stretched by $$0.15\ 	ext{m}. $$What is the magnitude of the force exerted by the spring?

Accepted Answer

$$F = kx = 50 	imes 0.15 = 7.5\ 	ext{N}$$

Question 3

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

Accepted Answer

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

Question 4

A mass of $$0.25\ 	ext{kg} is $$attached to a spring with $$k = 10\ 	ext{N/m}. $$What is the period of oscillation?

Accepted Answer

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

Question 5

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. What is the speed of the bullet-block system after the collision?

Accepted Answer

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

Question 6

A mass-spring system has a total mechanical energy of $$0.5\ 	ext{J}$$ and an amplitude of $$0.2\ 	ext{m}. $$What is the spring constant?

Accepted Answer

$$E = \frac{1}{2} k A^2$ \Rightarrow k = \frac{2E}{A^2$} = \frac{2 	imes 0.5}{0.2^2$} = 25\ 	ext{N/m}$$

Question 7

At which location(s) is the kinetic energy of a mass-spring system a maximum?

Accepted Answer

$$x = 0$$

Question 8

A simple pendulum of length $$L is $$oscillating on Earth. Which of the following factors affect its period?

Accepted Answer

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

Question 9

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

Accepted Answer

$$E = \frac{1}{2} k A^2$ = \frac{1}{2} 	imes 200 	imes 3^2$ = 900\ 	ext{J}$$

Question 10

A mass-spring system with mass $$m = 0.5\ 	ext{kg}$$ and spring constant $$k = 120\ 	ext{N/m}$$ oscillates. What is its period?

Accepted Answer

$$T = 2\pi \sqrt{\frac{0.5}{120}} \approx 0.41\ 	ext{s}$$

Question 11

A simple pendulum of mass $$0.5\ 	ext{kg}$$ and length $$1.5\ 	ext{m}$$ reaches a maximum height of $$0.08\ 	ext{m}. $$What is the maximum gravitational potential energy?

Accepted Answer

$$U = mgh = 0.5 	imes 9.8 	imes 0.08 = 0.39\ 	ext{J}$$

Question 12

A mass-spring system with mass $$m = 1.4\ 	ext{kg}$$ and spring constant $$k = 75.0\ 	ext{N/m} is $$stretched by $$5.0\ 	ext{cm}. $$What is the maximum speed of the mass?

Accepted Answer

$$v_{	ext{max}} = \sqrt{\frac{k A^2$}{m}} = \sqrt{\frac{75 	imes 0.05^2$}{1.4}} \approx 0.32\ 	ext{m/s}$$

Question 13

A mass-spring system with mass $$m = 0.15\ 	ext{kg}$$ and spring constant $$k = 30.2\ 	ext{N/m}$$ oscillates. What is its period?

Accepted Answer

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

Question 14

A mass-spring system has an amplitude $$A = 0.4\ 	ext{m}$$ and period $$T = 2.0\ 	ext{s}. $$How far does the mass travel in $$4.0\ 	ext{s}$$?

Accepted Answer

$$	ext{Distance} = 2 	imes 4A = 2 	imes 4 	imes 0.4 = 3.2\ 	ext{m}$$

Question 15

A mass-spring system with mass $$m = 4.0\ 	ext{kg}$$ and spring constant $$k = 64\ 	ext{N/m}$$ oscillates. What is its frequency?

Accepted Answer

$$f = \frac{1}{2\pi} \sqrt{\frac{64}{4.0}} \approx 0.64\ 	ext{Hz}$$

Simple Harmonic Motion: Mass-Spring Systems and Pendulums

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