Q: A 100 g mass oscillates on a massless spring with position given by $$x = (3.9\ \mathrm{cm}) \cos\left(\dfrac{2\pi t}{0.38\ \mathrm{s}}\right). $$What is the angular frequency $$\omega of $$the oscillation?

$$\omega = 2\pi \times 2.63\ \mathrm{Hz} = 16.5\ \mathrm{rad/s}$$

Q: Given the same mass and spring as above, what is the spring constant $$k$$?

$$k = m\$$omega^2$$ = 0.1\ \mathrm{kg} \times (16.5\ \mathrm{rad/s})^2$ = 27.3\ \mathrm{N/m}$$

Q: For the oscillator described by $$x = (3.9\ \mathrm{cm}) \cos\left(\dfrac{2\pi t}{0.38\ \mathrm{s}}\right)$$, what is the total mechanical energy of the system?

$$E = \dfrac{1}{2}kA^2$ = \dfrac{1}{2} \times 27.3\ \mathrm{N/m} \times (0.039\ \mathrm{m})^2$ = 0.0208\ \mathrm{J} = 20.8\ \mathrm{mJ}$$

Q: For the oscillator $$x = (3.9\ \mathrm{cm}) \cos\left(\dfrac{2\pi t}{0.38\ \mathrm{s}}\right)$$, what is the speed of the object when $$x = 3.0\ \mathrm{cm}$$?

$$v = \omega \sqrt{A^2$ - x^2$} = 16.5 \times \sqrt{(3.9)^2$ - (3.0)^2$}\ \mathrm{cm/s} = 41.2\ \mathrm{cm/s}$$

Question 1

A 100 g mass oscillates on a massless spring with position given by $$x = (3.9\ \mathrm{cm}) \cos\left(\dfrac{2\pi t}{0.38\ \mathrm{s}}\right). $$What is the angular frequency $$\omega of $$the oscillation?

Accepted Answer

$$\omega = 2\pi 	imes 2.63\ \mathrm{Hz} = 16.5\ \mathrm{rad/s}$$

Question 2

Given the same mass and spring as above, what is the spring constant $$k$$?

Accepted Answer

$$k = m\$$omega^2$$ = 0.1\ \mathrm{kg} 	imes (16.5\ \mathrm{rad/s})^2$ = 27.3\ \mathrm{N/m}$$

Question 3

For the oscillator described by $$x = (3.9\ \mathrm{cm}) \cos\left(\dfrac{2\pi t}{0.38\ \mathrm{s}}\right)$$, what is the total mechanical energy of the system?

Accepted Answer

$$E = \dfrac{1}{2}kA^2$ = \dfrac{1}{2} 	imes 27.3\ \mathrm{N/m} 	imes (0.039\ \mathrm{m})^2$ = 0.0208\ \mathrm{J} = 20.8\ \mathrm{mJ}$$

Question 4

A 3.8 kg mass oscillates on a spring with $$x(t) = A \sin(\omega t)$$, where $$A = 0.50\ \mathrm{m}$$ and $$\omega = 1.57\ \mathrm{rad/s}. $$What is the period $$T of $$the oscillation?

Accepted Answer

$$T = \dfrac{2\pi}{\omega} = \dfrac{2\pi}{1.57} = 4.00\ \mathrm{s}$$

Question 5

For the mass in Question 2, when is the force on the object a maximum between $$t = 2\ \mathrm{s}$$ and $$t = 4\ \mathrm{s}$$?

Accepted Answer

$$t = 3\ \mathrm{s}$$

Question 6

A 0.95 kg mass attached to a vertical spring of force constant $$k = 130\ \mathrm{N/m}$$ oscillates with a maximum speed of $$0.45\ \mathrm{m/s}. $$What is the amplitude $$A of $$the motion?

Accepted Answer

$$A = \dfrac{v_{max}}{\omega} = \dfrac{0.45}{\sqrt{130/0.95}} = 0.0385\ \mathrm{m} = 3.85\ \mathrm{cm}$$

Question 7

A student exploring a cave uses a rock on a rope as a pendulum to measure the depth of a hole. The rock takes 233 seconds to complete 10 oscillations. What is the depth of the hole?

Accepted Answer

$$L = \left(\dfrac{T}{2\pi}\$$right)^2$$ g = \left(\dfrac{23.3}{2\pi}\$$right)^2$$ 	imes 9.8 = 135\ \mathrm{m}$$

Question 8

For the oscillator $$x = (3.9\ \mathrm{cm}) \cos\left(\dfrac{2\pi t}{0.38\ \mathrm{s}}\right)$$, what is the speed of the object when $$x = 3.0\ \mathrm{cm}$$?

Accepted Answer

$$v = \omega \sqrt{A^2$ - x^2$} = 16.5 	imes \sqrt{(3.9)^2$ - (3.0)^2$}\ \mathrm{cm/s} = 41.2\ \mathrm{cm/s}$$

Question 9

For the same oscillator, does the speed at $$x = 3.0\ \mathrm{cm}$$ depend on the direction of motion?

Accepted Answer

No, because speed is the magnitude of velocity and is independent of direction.

Question 10

For the oscillator in Question 1, what is the sum of the kinetic and potential energy at $$x = 3.0\ \mathrm{cm}$$?

Accepted Answer

$$E_{total} = KE + PE = 8.49\ \mathrm{mJ} + 12.3\ \mathrm{mJ} = 20.8\ \mathrm{mJ}$$

Question 11

Why is the total energy at $$x = 3.0\ \mathrm{cm}$$ equal to the total energy calculated for the oscillator?

Accepted Answer

Because the total mechanical energy in simple harmonic motion is conserved.

Question 12

A 0.95 kg mass on a spring with $$k = 130\ \mathrm{N/m}$$ has a period $$T. $$What is $$T$$?

Accepted Answer

$$T = 2\pi \sqrt{\dfrac{m}{k}} = 2\pi \sqrt{\dfrac{0.95}{130}} = 0.537\ \mathrm{s}$$

Question 13

For the mass in Question 3, what is the maximum magnitude of acceleration?

Accepted Answer

$$a_{max} = \$$omega^2$$ A = \left(\sqrt{\dfrac{130}{0.95}}\$$right)^2$$ 	imes 0.0385 = 5.26\ \mathrm{m/s^2$}$$

Question 14

For the mass oscillating with $$x(t) = 0.50 \sin(1.57 t)$$, at what time is the mass first at $$x = 0.35\ \mathrm{m}$$?

Accepted Answer

$$t = \dfrac{\arcsin(0.35/0.50)}{1.57} = 0.494\ \mathrm{s}$$

Question 15

For the same mass, when is the mass at $$x = 0.35\ \mathrm{m}$$ for the second time?

Accepted Answer

$$t = \dfrac{\pi - \arcsin(0.35/0.50)}{1.57} = 1.51\ \mathrm{s}$$

Question 16

For the mass in Question 2, when the force on the object is a minimum, is the speed of the object a minimum or maximum?

Accepted Answer

Maximum, because force is zero at equilibrium where speed is greatest.

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