Oscillations & Gravitation Practice Test

1.

A mass \(m\) attached to a spring with spring constant \(k\) oscillates on a frictionless surface. If the amplitude is doubled, by what factor does the period change?

a) It doubles. b) It quadruples. c) It remains the same. d) It increases by a factor of \(\sqrt{2}\).

2.

An object undergoing SHM has position \(x(t) = 0.2\cos(4t)\). What is the phase at \(t=0.5\) seconds?

a) \(2\) radians b) \(4\) radians c) \(6\) radians d) \(8\) radians

3.

A 0.5-kg mass oscillates with frequency 2 Hz on a spring. What is the maximum kinetic energy during the motion if the amplitude is 0.1 m?

a) \(0.2\text{ J}\) b) \(0.4\text{ J}\) c) \(0.8\text{ J}\) d) \(1.6\text{ J}\)

4.

A pendulum of length 2 m oscillates with a maximum angular displacement of \(10^\circ\). What is the maximum speed of the pendulum?

a) \(0.45 \text{ m/s}\) b) \(0.77\text{ m/s}\) c) \(0.90\text{ m/s}\) d) \(0.55\text{ m/s}\)

5.

Two masses \(m\) and \(2m\) interact gravitationally. If their separation is reduced to \(\frac{1}{3}\) of the original distance, by what factor does the gravitational potential energy change?

a) It doubles. b) It stays the same. c) It becomes 3 times more negative. d) It becomes 3 times less negative.

6.

The gravitational potential energy between \(m_1\) and \(m_2\) separated by distance \(r\) is \(U = -\frac{G m_1 m_2}{r}\). What is the work required to move the masses from separation \(r\) to \(2r\)?

a) \(-\frac{G m_1 m_2}{2r}\) b) \(\frac{G m_1 m_2}{2r}\) c) \(\frac{G m_1 m_2}{r}\) d) \(\frac{G m_1 m_2}{r^2}\)

7.

A mass-spring system oscillates with amplitude \(A\) and spring constant \(k\). At displacement \(x = \frac{A}{2}\), what fraction of the total mechanical energy is kinetic?

a) \(\frac{3}{4}\) b) \(\frac{1}{2}\) c) \(\frac{1}{4}\) d) \(\frac{7}{8}\)

8.

A 1-kg mass attached to a spring (\(k=100\text{ N/m}\)) oscillates with amplitude \(0.1\text{ m}\). What is the total energy of the system, and what is the speed at displacement \(x=0.05\text{ m}\)?

a) \(0.5\text{ J},\ .87\text{ m/s}\) b) \(0.5\text{ J},\ .50\text{ m/s}\) c) \(0.25\text{ J},\ 2.74\text{ m/s}\) d) \(0.25\text{ J},\ .50\text{ m/s}\)

9.

A planet of mass \(m\) orbits a star of mass \(M\) in a circular orbit of radius \(r\). If the radius is halved, by what factor does the orbital speed change?

a) Increases by \(\sqrt{2}\) b) Increases by \(2\) c) Increases by \(\sqrt{4}\) d) Decreases by \(\sqrt{2}\)

10.

Given \(x(t) = A\cos(\omega t)\), what is the maximum jerk (time derivative of acceleration) magnitude?

a) \(A\omega\) b) \(A\omega^2\) c) \(A\omega^3\) d) \(A\omega^4\)

11.

A spring with spring constant \(k\) is stretched by a distance \(x\). How much work is done to stretch the spring from 0 to \(x\)?

a) \(kx\) b) \(\frac{1}{2}kx^2\) c) \(\frac{1}{2}kx\) d) \(kx^2\)

12.

A mass oscillates on a spring with angular frequency \(\omega\). If the mass is quadrupled, how does the angular frequency change?

a) It doubles. b) It halves. c) It quadruples. d) It remains the same.

13.

The period of a simple pendulum is \(T = 2\pi \sqrt{\frac{L}{g}}\). If the acceleration due to gravity decreases by 10%, how does the period change?

a) Increases by about 5% b) Decreases by about 5% c) Increases by about 10% d) Decreases by about 10%

14.

A mass oscillates horizontally with amplitude \(A\) and angular frequency \(\omega\). Write the expression for the total mechanical energy and determine its units.

a) \(E = \frac{1}{2} m \omega^2 A^2\) b) \(E = \frac{1}{2} k A^2\) c) Both a and b are correct and equivalent d) \(E = m g A\)

15.

A damped oscillator experiences a force proportional to velocity: \(F_d = -b v\). How does increasing \(b\) affect the system's oscillation frequency?

a) The frequency increases. b) The frequency decreases. c) The frequency remains the same. d) The frequency first increases then decreases.

16.

A 3 kg block hangs vertically from a spring with spring constant 150 N/m. What is the displacement of the block from the spring’s natural length at the equilibrium position?

a) 0.1 m
b) 0.15 m
c) 0.25 m
d) 0.2 m

17.

For a planet orbiting a star in an elliptical orbit, which parameter determines the orbital period?

a) The semi-major axis length b) The eccentricity c) The mass of the planet d) The distance at perihelion

18.

The restoring force on a mass attached to a spring is \(F = -kx\). What type of potential energy function corresponds to this force?

a) Linear potential energy b) Quadratic potential energy c) Cubic potential energy d) Constant potential energy

19.

Which expression represents the maximum velocity of a mass-spring system with amplitude \(A\) and angular frequency \(\omega\)?

a) \(A\omega\) b) \(A/\omega\) c) \(\frac{1}{2} A \omega^2\) d) \(\frac{1}{2} A \omega\)

20.

A block on a frictionless surface attached to a spring oscillates with total mechanical energy \(E\). What happens to \(E\) if the amplitude doubles?

a) \(E\) doubles b) \(E\) quadruples c) \(E\) halves d) \(E\) remains the same

21.

A mass \(m=2\,\text{kg}\) oscillates on a spring with \(k=50\,\text{N/m}\). What is the period of oscillation?

a) \(0.63\,\text{s}\) b) \(0.89\,\text{s}\) c) \(1.26\,\text{s}\) d) \(1.41\,\text{s}\)

22.

A masses \(m_1 = 3\,\text{kg}\) is connected to a spring and oscillates with frequency 2 Hz. What is the effective spring constant \(k\)?

a) \(118\,\text{N/m}\) b) \(237\,\text{N/m}\) c) \(474\,\text{N/m}\) d) \(948\,\text{N/m}\)

23.

A planet orbits a star with mass \(M = 2 \times 10^{30}\,\text{kg}\) at a radius \(r = 1.5 \times 10^{11}\,\text{m}\). Calculate the orbital speed \(v\) of the planet. (Use \(G = 6.67 \times 10^{-11}\,\text{N}\cdot\text{m}^2/\text{kg}^2\))

a) \(2.98 \times 10^4\,\text{m/s}\) b) \(3.50 \times 10^4\,\text{m/s}\) c) \(1.00 \times 10^5\,\text{m/s}\) d) \(4.00 \times 10^4\,\text{m/s}\)

24.

Two satellites orbit Earth at altitudes \(h_1\) and \(h_2\) such that \(h_2 = 4h_1\). If the orbital period of the lower satellite is \(T_1\), what is the period \(T_2\) of the higher satellite?

a) \(2T_1\) b) \(4T_1\) c) \(8T_1\) d) \(16T_1\)

25.

The gravitational force between two masses is \(F = 100\,\text{N}\) at separation \(r\). What is the force when the distance is tripled?

a) \(33.3\,\text{N}\) b) \(11.1\,\text{N}\) c) \(3.7\,\text{N}\) d) \(300\,\text{N}\)

26.

Calculate the escape velocity from a planet of mass \(5 \times 10^{24}\,\text{kg}\) and radius \(6.4 \times 10^6\,\text{m}\). (Use \(G=6.67 \times 10^{-11}\))

a) \(7.9 \times 10^3\,\text{m/s}\) b) \(1.1 \times 10^4\,\text{m/s}\) c) \(6.5 \times 10^3\,\text{m/s}\) d) \(9.8 \times 10^3\,\text{m/s}\)

27.

A mass \(0.5\,\text{kg}\) oscillates on a spring with amplitude \(0.1\,\text{m}\) and spring constant \(200\,\text{N/m}\). Calculate the maximum speed.

a) \(2.0\,\text{m/s}\) b) \(4.0\,\text{m/s}\) c) \(1.0\,\text{m/s}\) d) \(0.5\,\text{m/s}\)

28.

How much potential energy is stored in a spring stretched by \(0.05\,\text{m}\) with spring constant \(400\,\text{N/m}\)?

a) \(0.25\,\text{J}\) b) \(0.50\,\text{J}\) c) \(1.0\,\text{J}\) d) \(2.0\,\text{J}\)

29.

Calculate the gravitational potential energy between two \(1000\,\text{kg}\) masses separated by \(5\,\text{m}\). (Use \(G=6.67 \times 10^{-11}\))

a) \(-1.33 \times 10^{-5}\, \text{J}\) b) \(-1.33 \times 10^{-4}\, \text{J}\) c) \(-6.67 \times 10^{-6}\, \text{J}\) d) \(-6.67 \times 10^{-5}\, \text{J}\)

30.

Calculate the acceleration due to gravity on the surface of a moon with radius \(1.7 \times 10^6\,\text{m}\) and mass \(7.3 \times 10^{22}\,\text{kg}\). (Use \(G=6.67 \times 10^{-11}\))

a) \(9.8\, \text{m/s}^2\) b) \(0.82\, \text{m/s}^2\) c) \(3.7\, \text{m/s}^2\) d) \(1.62\, \text{m/s}^2\)

Oscillations & Gravitation Practice Test

10 Questions – Mixed Conceptual & Calculations