Oscillations Chapter-Wise Test 17

Correct answer Carries: 4.

Wrong Answer Carries: -1.

A mass of \( 4 \, \text{kg} \) is attached to a spring with \( k = 1600 \, \text{N/m} \) and displaced by \( 5 \, \text{cm} \). What is the total energy?

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

\( A = 0.05 \, \text{m}, k = 1600 \, \text{N/m} \).

\( E = 0.5 \times 1600 \times (0.05)^2 = 0.5 \times 1600 \times 0.0025 = 2 \, \text{J} \).

1.5 J
2.0 J
2.5 J
3.0 J
2

What happens to the period of a spring-mass system if both the mass and spring constant are doubled?

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

It doubles
It remains unchanged
It increases by \( \sqrt{2} \)
It halves
2

A spring-mass system oscillates with \( T = 0.4 \, \text{s} \) when \( m = 0.2 \, \text{kg} \). What is the spring constant?

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

\( 0.4 = 2\pi \sqrt{\frac{0.2}{k}} \Rightarrow \frac{0.4}{2\pi} = \sqrt{\frac{0.2}{k}} \).

\( (0.0637)^2 = \frac{0.2}{k} \Rightarrow k = \frac{0.2}{0.00406} \approx 49.26 \, \text{N/m} \).

40 N/m
45 N/m
49.26 N/m
50 N/m
3

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

(a) \( 2 \cos (\omega t) \): SHM.

(b) \( \cos \omega t + \cos 3\omega t \): Periodic (period \( \frac{2\pi}{\omega} \)), not SHM (multiple frequencies).

(c) \( 3 \sin (2\omega t) \): SHM.

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

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

A spring-mass system has \( m = 2 \, \text{kg}, k = 800 \, \text{N/m} \). What is its angular frequency?

\( \omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{800}{2}} = \sqrt{400} = 20 \, \text{rad/s} \).

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

Two identical springs (\( k = 100 \, \text{N/m} \)) are attached to a \( 2.0 \, \text{kg} \) mass as in Fig. 13.14. What is the period?

Effective \( k_{\text{eff}} = 2k = 2 \times 100 = 200 \, \text{N/m} \).

\( T = 2\pi \sqrt{\frac{m}{k_{\text{eff}}}} = 2\pi \sqrt{\frac{2}{200}} = 2\pi \sqrt{0.01} = 2\pi \times 0.1 \approx 0.628 \, \text{s} \).

0.5 s
0.6 s
0.62 s
0.628 s
4

Which statement correctly describes the relationship between SHM and uniform circular motion?

SHM is the one-dimensional projection of uniform circular motion along a diameter, with the same period but different force characteristics (linear vs. centripetal).

SHM requires a centripetal force like circular motion
SHM and circular motion have identical force laws
SHM is the projection of uniform circular motion on a diameter
SHM occurs in two dimensions like circular motion
3

A mass oscillates with \( v = -8 \cos (2t) \) (in m/s). What is its displacement function?

Velocity: \( v = -\omega A \sin (\omega t) \), but given \( v = -8 \cos (2t) \).

\( \omega = 2 \, \text{s}^{-1}, v_{\text{max}} = \omega A = 8 \Rightarrow A = \frac{8}{2} = 4 \, \text{m} \).

Since \( v = -A \omega \sin (\omega t) \), adjust phase: \( x = 4 \sin (2t) \).

\( 4 \cos (2t) \)
\( 4 \sin (2t) \)
\( 8 \sin (2t) \)
\( 8 \cos (2t) \)
2

A spring of \( k = 200 \, \text{N/m} \) has a \( 0.5 \, \text{kg} \) mass. If \( E = 1 \, \text{J} \), what is the amplitude?

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

\( 1 = \frac{1}{2} \times 200 \times A^2 \Rightarrow 1 = 100 A^2 \Rightarrow A^2 = 0.01 \Rightarrow A = 0.1 \, \text{m} \).

0.05 m
0.08 m
0.12 m
0.1 m
4

A mass of \( 0.5 \, \text{kg} \) on a spring has \( E = 2 \, \text{J} \) at \( A = 20 \, \text{cm} \). What is the spring constant?

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

\( 2 = \frac{1}{2} k (0.2)^2 \Rightarrow 2 = 0.02 k \Rightarrow k = \frac{2}{0.02} = 100 \, \text{N/m} \).

50 N/m
75 N/m
100 N/m
125 N/m
3

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