Oscillations Chapter-Wise Test 12

Correct answer Carries: 4.

Wrong Answer Carries: -1.

A particle’s displacement is \( x = 5 \sin (\pi t + \frac{\pi}{3}) \) (in m). What is its velocity at \( t = 0 \, \text{s} \)? (Take \( \cos \frac{\pi}{3} = 0.5 \))

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

\( A = 5 \, \text{m}, \omega = \pi \, \text{s}^{-1}, \phi = \frac{\pi}{3} \).

At \( t = 0 \): \( v = \pi \times 5 \cos (\frac{\pi}{3}) = 5\pi \times 0.5 = 2.5\pi \approx 7.85 \, \text{m/s} \).

7.85 m/s
5.0 m/s
10.0 m/s
12.0 m/s
1

A pendulum of length \( 2 \, \text{m} \) oscillates with \( g = 10 \, \text{m/s}^2 \). What is its frequency?

Period: \( T = 2\pi \sqrt{\frac{L}{g}} = 2\pi \sqrt{\frac{2}{10}} = 2\pi \sqrt{0.2} \approx 2.8 \, \text{s} \).

Frequency: \( v = \frac{1}{T} = \frac{1}{2.8} \approx 0.357 \, \text{Hz} \).

0.25 Hz
0.30 Hz
0.357 Hz
0.40 Hz
3

A pendulum oscillates with \( \theta_{\text{max}} = 0.2 \, \text{rad}, L = 2 \, \text{m}, g = 10 \, \text{m/s}^2 \). What is its maximum speed?

\( \omega = \sqrt{\frac{g}{L}} = \sqrt{\frac{10}{2}} = \sqrt{5} \approx 2.24 \, \text{rad/s} \).

Arc length amplitude: \( A = L \theta_{\text{max}} = 2 \times 0.2 = 0.4 \, \text{m} \).

\( v_{\text{max}} = \omega A = 2.24 \times 0.4 \approx 0.896 \, \text{m/s} \).

0.6 m/s
0.8 m/s
0.896 m/s
1.0 m/s
3

What effect does tripling the spring constant have on the period of a spring-mass system if the mass is also tripled?

Period \( T = 2\pi \sqrt{\frac{m}{k}} \). If \( k' = 3k \) and \( m' = 3m \), then \( T' = 2\pi \sqrt{\frac{3m}{3k}} = 2\pi \sqrt{\frac{m}{k}} = T \), so the period remains unchanged.

It triples
It remains unchanged
It increases by \( \sqrt{3} \)
It decreases to one-third
2

A spring-mass system has \( m = 0.25 \, \text{kg}, k = 100 \, \text{N/m} \). If displaced by \( 6 \, \text{cm} \), what is the total energy?

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

\( A = 0.06 \, \text{m}, k = 100 \, \text{N/m} \).

\( E = 0.5 \times 100 \times (0.06)^2 = 0.5 \times 100 \times 0.0036 = 0.18 \, \text{J} \).

0.18 J
0.2 J
0.25 J
0.3 J
1

A particle in SHM has \( x = 7 \sin (3t + \frac{\pi}{6}) \) (in m). What is its acceleration at \( t = 0 \, \text{s} \)? (Take \( \sin 30^\circ = 0.5 \))

Acceleration: \( a = -\omega^2 x \).

\( \omega = 3 \, \text{s}^{-1}, x(0) = 7 \sin \frac{\pi}{6} = 7 \times 0.5 = 3.5 \, \text{m} \).

\( a = -3^2 \times 3.5 = -9 \times 3.5 = -31.5 \, \text{m/s}^2 \).

-28 m/s²
-30 m/s²
-31.5 m/s²
-35 m/s²
3

A particle in SHM has \( a = -9 x \) (in SI units). What is its period?

For SHM, \( a = -\omega^2 x \). Given \( a = -9 x \), \( \omega^2 = 9 \Rightarrow \omega = 3 \, \text{rad/s} \).

\( T = \frac{2\pi}{\omega} = \frac{2\pi}{3} \approx 2.09 \, \text{s} \).

1.5 s
1.8 s
2.09 s
2.5 s
3

A particle in SHM has an amplitude of \( 8 \, \text{cm} \) and a frequency of \( 2 \, \text{Hz} \). What is its maximum acceleration? (Take \( \pi = 3.14 \))

Maximum acceleration: \( a_{\text{max}} = \omega^2 A \).

\( \omega = 2\pi v = 2 \times 3.14 \times 2 = 12.56 \, \text{rad/s} \).

\( A = 8 \, \text{cm} = 0.08 \, \text{m} \).

\( a_{\text{max}} = (12.56)^2 \times 0.08 \approx 157.75 \times 0.08 \approx 12.62 \, \text{m/s}^2 \).

12.62 m/s²
10.0 m/s²
15.0 m/s²
18.0 m/s²
1

A spring-mass system has \( m = 0.6 \, \text{kg}, k = 240 \, \text{N/m} \). If displaced by \( 7 \, \text{cm} \), what is the total energy?

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

\( A = 0.07 \, \text{m}, k = 240 \, \text{N/m} \).

\( E = 0.5 \times 240 \times (0.07)^2 = 0.5 \times 240 \times 0.0049 = 0.588 \, \text{J} \).

0.4 J
0.5 J
0.588 J
0.7 J
3

A particle’s motion is given by \( x = 2 \cos (5t - \frac{\pi}{4}) \) (in m). What is its kinetic energy at \( x = 1 \, \text{m} \) if \( m = 1 \, \text{kg} \)?

Total energy: \( E = \frac{1}{2} k A^2 = \frac{1}{2} m \omega^2 A^2 = 0.5 \times 1 \times 5^2 \times 2^2 = 50 \, \text{J} \).

Potential energy: \( U = \frac{1}{2} m \omega^2 x^2 = 0.5 \times 1 \times 25 \times 1^2 = 12.5 \, \text{J} \).

Kinetic energy: \( K = E - U = 50 - 12.5 = 37.5 \, \text{J} \).

25.0 J
30.0 J
37.5 J
42.5 J
3

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