Oscillations Chapter-Wise Test 13

Correct answer Carries: 4.

Wrong Answer Carries: -1.

A spring-mass system oscillates with \( T = 0.5 \, \text{s} \) when \( m = 0.5 \, \text{kg} \). What is the spring constant?

\( T = 2\pi \sqrt{\frac{m}{k}} \).

\( 0.5 = 2\pi \sqrt{\frac{0.5}{k}} \Rightarrow \frac{0.5}{2\pi} = \sqrt{\frac{0.5}{k}} \).

\( (0.0796)^2 = \frac{0.5}{k} \Rightarrow k = \frac{0.5}{0.00634} \approx 78.9 \, \text{N/m} \).

50 N/m
60 N/m
78.9 N/m
100 N/m
3

What fundamental property of the restoring force distinguishes simple harmonic motion from other oscillatory motions?

In SHM, the restoring force must be directly proportional to displacement and opposite in direction (\( F = -kx \)), ensuring a linear relationship, unlike non-linear oscillatory systems.

Its linearity with displacement
Its dependence on velocity
Its constant magnitude
Its variation with time
1

In SHM, which quantity is directly proportional to the square of the angular frequency?

Acceleration \( a = -\omega^2 x \) has a maximum magnitude of \( \omega^2 A \), making it proportional to \( \omega^2 \), unlike velocity (\( \omega A \)) or displacement.

Displacement
Velocity
Kinetic energy
Acceleration
4

A spring-mass system has \( m = 0.8 \, \text{kg}, k = 320 \, \text{N/m} \). If displaced by \( 5 \, \text{cm} \), what is the total energy?

Total energy: \( E = \frac{1}{2} k A^2 \).

\( A = 0.05 \, \text{m}, k = 320 \, \text{N/m} \).

\( E = 0.5 \times 320 \times (0.05)^2 = 0.5 \times 320 \times 0.0025 = 0.4 \, \text{J} \).

0.3 J
0.4 J
0.5 J
0.6 J
2

A simple pendulum has a period of \( 2.5 \, \text{s} \) on Earth (\( g = 9.8 \, \text{m/s}^2 \)). What is its length?

\( T = 2\pi \sqrt{\frac{L}{g}} \).

\( 2.5 = 2\pi \sqrt{\frac{L}{9.8}} \Rightarrow \sqrt{\frac{L}{9.8}} = \frac{2.5}{2\pi} \approx 0.398 \).

\( \frac{L}{9.8} = (0.398)^2 \Rightarrow L \approx 9.8 \times 0.158 \approx 1.55 \, \text{m} \).

1.0 m
1.25 m
1.55 m
2.0 m
3

A pendulum has \( L = 1.2 \, \text{m}, g = 9.8 \, \text{m/s}^2 \). What is its angular frequency?

\( \omega = \sqrt{\frac{g}{L}} = \sqrt{\frac{9.8}{1.2}} \approx \sqrt{8.17} \approx 2.86 \, \text{rad/s} \).

2.0 rad/s
2.5 rad/s
2.86 rad/s
3.0 rad/s
3

A spring-mass system has \( m = 2.5 \, \text{kg}, k = 1000 \, \text{N/m} \). What is its angular frequency?

\( \omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{1000}{2.5}} = \sqrt{400} = 20 \, \text{rad/s} \).

15 rad/s
20 rad/s
25 rad/s
30 rad/s
2

A pendulum has \( L = 0.25 \, \text{m}, g = 10 \, \text{m/s}^2 \). What is its angular frequency?

\( \omega = \sqrt{\frac{g}{L}} = \sqrt{\frac{10}{0.25}} = \sqrt{40} \approx 6.32 \, \text{rad/s} \).

6.0 rad/s
6.2 rad/s
6.3 rad/s
6.32 rad/s
4

A mass oscillates with \( v = -12 \sin (6t) \) (in m/s). What is its amplitude?

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

\( \omega = 6 \, \text{s}^{-1}, v_{\text{max}} = \omega A = 12 \Rightarrow A = \frac{12}{6} = 2 \, \text{m} \).

1.5 m
2.0 m
2.5 m
3.0 m
2

A particle in SHM has \( x = 7 \sin (3\pi t + \frac{\pi}{3}) \) (in m). What is its speed at \( t = 0.5 \, \text{s} \)? (Take \( \cos 30^\circ = \frac{\sqrt{3}}{2} \))

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

\( A = 7 \, \text{m}, \omega = 3\pi \, \text{s}^{-1}, \phi = \frac{\pi}{3} \).

At \( t = 0.5 \): \( 3\pi \times 0.5 + \frac{\pi}{3} = \frac{3\pi}{2} + \frac{\pi}{3} = \frac{9\pi}{6} + \frac{2\pi}{6} = \frac{11\pi}{6} \).

\( v = 3\pi \times 7 \cos \frac{11\pi}{6} = 21\pi \cos (180^\circ - 30^\circ) = 21\pi \times \frac{\sqrt{3}}{2} \approx 57.07 \, \text{m/s} \).

55 m/s
56 m/s
57 m/s
57.07 m/s
4

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