Oscillations Chapter-Wise Test 1

Correct answer Carries: 4.

Wrong Answer Carries: -1.

Which condition ensures that the total mechanical energy in an SHM system remains conserved during the motion?

Total mechanical energy (kinetic + potential) is conserved in SHM when no external dissipative forces (e.g., friction) act, allowing energy to transform without loss.

Absence of dissipative forces
Constant amplitude
Variable angular frequency
Non-linear restoring force
1

A particle in SHM has \( x = 4 \cos (2t + \frac{\pi}{3}) \) (in m). What is its speed at \( t = 0.5 \, \text{s} \)? (Take \( \sin 60^\circ = \frac{\sqrt{3}}{2} \))

Velocity: \( v = -\omega A \sin (\omega t + \phi) \).

\( A = 4 \, \text{m}, \omega = 2 \, \text{s}^{-1}, \phi = \frac{\pi}{3} \).

At \( t = 0.5 \): \( 2 \times 0.5 + \frac{\pi}{3} = 1 + \frac{\pi}{3} \approx 2.047 \, \text{rad} \approx 117^\circ \).

\( v = -2 \times 4 \sin 117^\circ \approx -8 \sin (180^\circ - 63^\circ) \approx -8 \times 0.838 \approx -6.7 \, \text{m/s} \).

-6.0 m/s
-6.7 m/s
-7.0 m/s
-8.0 m/s
2

A pendulum has \( L = 0.6 \, \text{m}, g = 9.8 \, \text{m/s}^2 \). What is its angular frequency?

\( \omega = \sqrt{\frac{g}{L}} = \sqrt{\frac{9.8}{0.6}} \approx \sqrt{16.33} \approx 4.04 \, \text{rad/s} \).

3.5 rad/s
4.0 rad/s
4.04 rad/s
4.5 rad/s
3

A pendulum of length \( 0.36 \, \text{m} \) oscillates with \( g = 9.8 \, \text{m/s}^2 \). What is its frequency?

Period: \( T = 2\pi \sqrt{\frac{L}{g}} = 2\pi \sqrt{\frac{0.36}{9.8}} \approx 2 \times 3.14 \sqrt{0.0367} \approx 1.2 \, \text{s} \).

Frequency: \( v = \frac{1}{T} = \frac{1}{1.2} \approx 0.833 \, \text{Hz} \).

0.7 Hz
0.8 Hz
0.833 Hz
0.9 Hz
3

What underlies the periodic nature of SHM when expressed as a superposition of sine and cosine functions?

The periodicity arises from the repeating nature of sine and cosine functions, which have a fixed period (\( 2\pi/\omega \)), ensuring the motion repeats consistently.

The amplitude variation
The phase shift
The periodicity of trigonometric functions
The linear restoring force
3

A pendulum has \( L = 1.4 \, \text{m}, g = 9.8 \, \text{m/s}^2 \). What is its angular frequency?

\( \omega = \sqrt{\frac{g}{L}} = \sqrt{\frac{9.8}{1.4}} \approx \sqrt{7} \approx 2.65 \, \text{rad/s} \).

2.0 rad/s
2.5 rad/s
2.65 rad/s
3.0 rad/s
3

Which of the following periodic motions cannot be classified as oscillatory due to the absence of a fixed equilibrium point?

Rotational motion of a ceiling fan is periodic but not oscillatory, as it lacks a fixed equilibrium point about which it moves to-and-fro, unlike SHM examples.

A mass vibrating on a spring
Rotational motion of a ceiling fan
A pendulum swinging symmetrically
A tuning fork vibrating
2

A particle in SHM has \( a = -36 x \) (in SI units). What is its frequency?

For SHM, \( a = -\omega^2 x \). Given \( a = -36 x \), \( \omega^2 = 36 \Rightarrow \omega = 6 \, \text{rad/s} \).

Frequency: \( v = \frac{\omega}{2\pi} = \frac{6}{2 \times 3.14} \approx 0.955 \, \text{Hz} \).

0.5 Hz
0.75 Hz
0.955 Hz
1.2 Hz
3

A particle’s motion is \( x = 4 \sin (3t - \frac{\pi}{3}) \) (in m). What is its velocity at \( t = \frac{\pi}{6} \, \text{s} \)? (Take \( \cos 30^\circ = \frac{\sqrt{3}}{2} \))

Velocity: \( v = \omega A \cos (\omega t + \phi) \).

\( A = 4 \, \text{m}, \omega = 3 \, \text{s}^{-1}, \phi = -\frac{\pi}{3} \).

At \( t = \frac{\pi}{6} \): \( 3 \times \frac{\pi}{6} - \frac{\pi}{3} = \frac{\pi}{2} - \frac{\pi}{3} = \frac{\pi}{6} \).

\( v = 3 \times 4 \cos \frac{\pi}{6} = 12 \times \frac{\sqrt{3}}{2} = 6\sqrt{3} \approx 10.39 \, \text{m/s} \).

8.0 m/s
9.0 m/s
10.39 m/s
12.0 m/s
3

What is the significance of the phase constant in the displacement equation of SHM?

The phase constant (\( \phi \) in \( x = A \cos (\omega t + \phi) \)) determines the initial position and velocity, setting the starting point of the oscillation cycle.

It fixes the initial position of the particle
It determines the amplitude of oscillation
It sets the frequency of the motion
It indicates the maximum velocity
1

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