Oscillations Chapter-Wise Test 14

Correct answer Carries: 4.

Wrong Answer Carries: -1.

A particle’s x-projection from circular motion is \( x = 10 \cos (2\pi t) \) (in m). What is its maximum acceleration?

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

\( A = 10 \, \text{m}, \omega = 2\pi \, \text{s}^{-1} \).

\( a_{\text{max}} = (2\pi)^2 \times 10 \approx 39.48 \times 10 \approx 394.8 \, \text{m/s}^2 \).

350 m/s²
394.8 m/s²
400 m/s²
450 m/s²
2

A spring system has \( k = 800 \, \text{N/m}, m = 2 \, \text{kg} \). What is the maximum speed if amplitude is \( 5 \, \text{cm} \)?

\( \omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{800}{2}} = 20 \, \text{rad/s} \).

\( v_{\text{max}} = \omega A = 20 \times 0.05 = 1.0 \, \text{m/s} \).

0.8 m/s
1.0 m/s
1.2 m/s
1.5 m/s
2

What critical condition must the acceleration satisfy for a motion to be classified as simple harmonic?

In SHM, acceleration must be proportional to displacement and directed opposite to it (\( a = -\omega^2 x \)), ensuring harmonic oscillation about the mean position.

It is proportional to displacement and opposite in direction
It is constant in magnitude and direction
It varies quadratically with velocity
It is independent of position
1

In SHM, what condition results in the potential energy being zero?

Potential energy \( U = \frac{1}{2} k x^2 \) is zero when displacement \( x = 0 \), which occurs at the mean position, where all energy is kinetic.

Maximum velocity
Maximum acceleration
Zero displacement
Maximum displacement
3

A spring-mass system has \( m = 1.6 \, \text{kg}, k = 640 \, \text{N/m} \). What is its angular frequency?

\( \omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{640}{1.6}} = \sqrt{400} = 20 \, \text{rad/s} \).

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

Two identical springs (\( k = 80 \, \text{N/m} \)) are attached to a \( 2 \, \text{kg} \) mass as in Fig. 13.14. What is the period?

Effective \( k_{\text{eff}} = 2k = 2 \times 80 = 160 \, \text{N/m} \).

\( T = 2\pi \sqrt{\frac{m}{k_{\text{eff}}}} = 2\pi \sqrt{\frac{2}{160}} = 2\pi \sqrt{0.0125} \approx 0.702 \, \text{s} \).

0.5 s
0.6 s
0.702 s
0.8 s
3

A spring system has \( m = 0.3 \, \text{kg}, k = 120 \, \text{N/m}, A = 8 \, \text{cm} \). What is the potential energy at \( x = 4 \, \text{cm} \)?

Potential energy: \( U = \frac{1}{2} k x^2 \).

\( k = 120 \, \text{N/m}, x = 0.04 \, \text{m} \).

\( U = 0.5 \times 120 \times (0.04)^2 = 0.5 \times 120 \times 0.0016 = 0.096 \, \text{J} \).

0.05 J
0.08 J
0.096 J
0.1 J
3

Which of the following is a non-periodic motion? (\( \omega \) is a positive constant)

(a) \( \sin \omega t + \cos \omega t \): Periodic, period \( \frac{2\pi}{\omega} \).

(b) \( \sin^3 \omega t \): Periodic, period \( \frac{2\pi}{3\omega} \).

(c) \( e^{-\omega t} \): Non-periodic (decays to zero).

(d) \( \cos 2\omega t \): Periodic, period \( \frac{\pi}{\omega} \).

\( \sin \omega t + \cos \omega t \)
\( \sin^3 \omega t \)
\( e^{-\omega t} \)
\( \cos 2\omega t \)
3

Two identical springs (\( k = 50 \, \text{N/m} \)) are attached to a \( 0.5 \, \text{kg} \) mass as in Fig. 13.14. What is the period?

Effective \( k_{\text{eff}} = 2k = 2 \times 50 = 100 \, \text{N/m} \).

\( T = 2\pi \sqrt{\frac{m}{k_{\text{eff}}}} = 2\pi \sqrt{\frac{0.5}{100}} = 2\pi \sqrt{0.005} \approx 0.44 \, \text{s} \).

0.3 s
0.4 s
0.44 s
0.5 s
3

Which condition must be satisfied for the potential energy in an SHM system to be expressible as a quadratic function of displacement?

Potential energy in SHM (\( U = \frac{1}{2} k x^2 \)) is quadratic when the force is conservative and linear (\( F = -kx \)), distinguishing SHM from non-linear systems.

The restoring force must be linear
The motion must be non-periodic
The force must vary with time
The amplitude must be constant
1

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