Oscillations Chapter-Wise Test 6

Correct answer Carries: 4.

Wrong Answer Carries: -1.

A mass of \( 2 \, \text{kg} \) on a spring has \( k = 800 \, \text{N/m} \) and \( A = 10 \, \text{cm} \). What is the kinetic energy at \( x = 5 \, \text{cm} \)?

Total energy: \( E = \frac{1}{2} k A^2 = 0.5 \times 800 \times (0.1)^2 = 4 \, \text{J} \).

Potential energy: \( U = \frac{1}{2} k x^2 = 0.5 \times 800 \times (0.05)^2 = 1 \, \text{J} \).

Kinetic energy: \( K = E - U = 4 - 1 = 3 \, \text{J} \).

2.0 J
3.0 J
3.5 J
4.0 J
2

A particle in SHM has an amplitude of \( 6 \, \text{cm} \) and a frequency of \( 3 \, \text{Hz} \). What is its maximum velocity? (Take \( \pi = 3.14 \))

Maximum velocity: \( v_{\text{max}} = A \omega \).

\( \omega = 2\pi v = 2 \times 3.14 \times 3 = 18.84 \, \text{rad/s} \).

\( A = 0.06 \, \text{m} \).

\( v_{\text{max}} = 0.06 \times 18.84 = 1.1304 \, \text{m/s} \).

1.1304 m/s
1.2 m/s
1.5 m/s
1.8 m/s
1

A simple pendulum has a length of \( 1.6 \, \text{m} \) and oscillates with \( g = 9.8 \, \text{m/s}^2 \). What is its period?

Period: \( T = 2\pi \sqrt{\frac{L}{g}} = 2\pi \sqrt{\frac{1.6}{9.8}} \approx 2 \times 3.14 \sqrt{0.1633} \approx 2.54 \, \text{s} \).

2.0 s
2.3 s
2.54 s
2.8 s
3

A mass oscillates with \( v = -10 \sin (5t) \) (in m/s). What is its amplitude?

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

\( \omega = 5 \, \text{s}^{-1}, v_{\text{max}} = \omega A = 10 \Rightarrow A = \frac{10}{5} = 2 \, \text{m} \).

1.5 m
1.8 m
2.0 m
2.5 m
3

A simple pendulum has a length of \( 0.81 \, \text{m} \) and oscillates with \( g = 9.8 \, \text{m/s}^2 \). What is its period?

Period: \( T = 2\pi \sqrt{\frac{L}{g}} = 2\pi \sqrt{\frac{0.81}{9.8}} \approx 2 \times 3.14 \sqrt{0.0827} \approx 1.805 \, \text{s} \).

1.805 s
1.9 s
2.0 s
2.2 s
1

A particle’s displacement is \( x = 8 \sin (2\pi t - \frac{\pi}{3}) \) (in m). What is its velocity at \( t = 0.25 \, \text{s} \)? (Take \( \cos 30^\circ = \frac{\sqrt{3}}{2} \))

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

\( A = 8 \, \text{m}, \omega = 2\pi \, \text{s}^{-1}, \phi = -\frac{\pi}{3} \).

At \( t = 0.25 \): \( 2\pi \times 0.25 - \frac{\pi}{3} = \frac{\pi}{2} - \frac{\pi}{3} = \frac{\pi}{6} \).

\( v = 2\pi \times 8 \cos \frac{\pi}{6} = 16\pi \times \frac{\sqrt{3}}{2} \approx 43.54 \, \text{m/s} \).

40.0 m/s
42.0 m/s
43.54 m/s
45.0 m/s
3

A spring system has \( m = 1.2 \, \text{kg}, k = 480 \, \text{N/m}, A = 6 \, \text{cm} \). What is the potential energy at \( x = 3 \, \text{cm} \)?

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

\( k = 480 \, \text{N/m}, x = 0.03 \, \text{m} \).

\( U = 0.5 \times 480 \times (0.03)^2 = 0.5 \times 480 \times 0.0009 = 0.216 \, \text{J} \).

0.15 J
0.2 J
0.216 J
0.25 J
3

Which function represents SHM? (\( \omega \) is a positive constant)

SHM requires \( a = -\omega^2 x \):

(a) \( 5 \cos (3\omega t + \frac{\pi}{6}) \): \( a = -5 (3\omega)^2 \cos (3\omega t + \frac{\pi}{6}) = -\omega^2 x \), SHM.

(b) \( \sin \omega t + \cos 2\omega t \): Not SHM (mixed frequencies).

\( e^{-\omega t} \): Not periodic.

(d) \( \sin^2 \omega t \): Periodic, not SHM.

\( 5 \cos (3\omega t + \frac{\pi}{6}) \)
\( \sin \omega t + \cos 2\omega t \)
\( e^{-\omega t} \)
\( \sin^2 \omega t \)
1

Which of the following represents periodic motion but not SHM? (\( \omega \) is a positive constant)

(a) \( 4 \cos (\omega t) \): SHM.

(b) \( \sin \omega t + \sin 2\omega t \): Periodic (period \( \frac{2\pi}{\omega} \)), not SHM (multiple frequencies).

(c) \( 3 \sin (\omega t - \frac{\pi}{3}) \): SHM.

(d) \( e^{\omega t} \): Not periodic.

\( 4 \cos (\omega t) \)
\( \sin \omega t + \sin 2\omega t \)
\( 3 \sin (\omega t - \frac{\pi}{3}) \)
\( e^{\omega t} \)
2

What physical property of a simple pendulum primarily governs its oscillatory behavior?

The length of the pendulum determines the period (\( T = 2\pi \sqrt{\frac{L}{g}} \)), controlling the frequency and thus the oscillatory behavior, more than mass or amplitude for small angles.

The mass of the bob
The length of the string
The amplitude of oscillation
The material of the bob
2

What is the primary reason a simple pendulum’s period increases at higher altitudes?

The period \( T = 2\pi \sqrt{\frac{L}{g}} \) increases as \( g \) (acceleration due to gravity) decreases with altitude, inversely affecting the period.

The length of the pendulum increases
The mass of the bob increases
The amplitude decreases
The acceleration due to gravity decreases
4

A mass of \( 0.8 \, \text{kg} \) on a spring with \( k = 200 \, \text{N/m} \) has \( A = 10 \, \text{cm} \). What is the kinetic energy at \( x = 5 \, \text{cm} \)?

Total energy: \( E = \frac{1}{2} k A^2 = 0.5 \times 200 \times (0.1)^2 = 1 \, \text{J} \).

Potential energy: \( U = \frac{1}{2} k x^2 = 0.5 \times 200 \times (0.05)^2 = 0.25 \, \text{J} \).

Kinetic energy: \( K = E - U = 1 - 0.25 = 0.75 \, \text{J} \).

0.5 J
0.75 J
0.8 J
1.0 J
2

A spring system has \( m = 0.4 \, \text{kg}, k = 160 \, \text{N/m}, A = 7 \, \text{cm} \). What is the potential energy at \( x = 3.5 \, \text{cm} \)?

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

\( k = 160 \, \text{N/m}, x = 0.035 \, \text{m} \).

\( U = 0.5 \times 160 \times (0.035)^2 = 0.5 \times 160 \times 0.001225 = 0.098 \, \text{J} \).

0.08 J
0.09 J
0.098 J
0.1 J
3

A particle’s x-projection from circular motion is \( x = 5 \cos (2t) \) (in m). What is its maximum speed?

Maximum speed: \( v_{\text{max}} = \omega A \).

\( A = 5 \, \text{m}, \omega = 2 \, \text{s}^{-1} \).

\( v_{\text{max}} = 2 \times 5 = 10 \, \text{m/s} \).

5 m/s
8 m/s
10 m/s
12 m/s
3

In SHM, which quantity is maximized when the particle’s acceleration is at its peak?

Acceleration peaks at \( a_{\text{max}} = \omega^2 A \) when \( x = \pm A \) (extreme positions), where potential energy (\( U = \frac{1}{2} k A^2 \)) is maximum.

Kinetic energy
Velocity
Displacement
Potential energy
4

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