NEET Oscillations — Set 4

Question 1

A particle in SHM has $ v = -10 \sin (2t + \frac{\pi}{2}) $ (in m/s). What is its displacement function?

Accepted Answer

Velocity: $ v = -\omega A \sin (\omega t + \phi) $. Given $ \omega = 2, v_{\text{max}} = \omega A = 10 \Rightarrow A = 5 \, \text{m} $. Comparing: $ -\omega A \sin (2t + \frac{\pi}{2}) = -10 \sin (2t + \frac{\pi}{2}) $, phase matches. Displacement: $ x = A \cos (\omega t + \phi) = 5 \cos (2t + \frac{\pi}{2}) $.

Question 2

Which property of an SHM system ensures that the frequency remains constant regardless of the energy imparted?

Accepted Answer

Frequency ($ v = \frac{1}{2\pi} \sqrt{\frac{k}{m}} $) depends solely on mass and spring constant, not energy or amplitude, due to the linear nature of the restoring force.

Question 3

A particle in SHM has $ x = 5 \sin (3t + \frac{\pi}{4}) $ (in m). What is its speed at $ t = 0.25 \, \text{s} $? (Take $ \cos 45^\circ = \frac{\sqrt{2}}{2} $)

Accepted Answer

Velocity: $ v = \omega A \cos (\omega t + \phi) $. $ A = 5 \, \text{m}, \omega = 3 \, \text{s}^{-1}, \phi = \frac{\pi}{4} $. At $ t = 0.25 $: $ 3 \times 0.25 + \frac{\pi}{4} = 0.75 + 0.785 \approx 1.535 \, \text{rad} \approx 88^\circ $. $ v = 3 \times 5 \cos 88^\circ \approx 15 \times 0.0349 \approx 0.5235 \, \text{m/s} $.

NEET Physics: Oscillations — Practice Set 4

Q1. A particle in SHM has \( v = -10 \sin (2t + \frac{\pi}{2}) \) (in m/s). What is its displacement function?

Q2. Which property of an SHM system ensures that the frequency remains constant regardless of the energy imparted?

Q3. A particle in SHM has \( x = 5 \sin (3t + \frac{\pi}{4}) \) (in m). What is its speed at \( t = 0.25 \, \text{s} \)? (Take \( \cos 45^\circ = \frac{\sqrt{2}}{2} \))

Q4. Which function represents SHM? (\( \omega \) is a positive constant)

Q5. A particle in SHM has an amplitude of \( 15 \, \text{cm} \) and a frequency of \( 2 \, \text{Hz} \). What is its maximum velocity? (Take \( \pi = 3.14 \))

Q6. Two springs (\( k = 150 \, \text{N/m} \)) are attached to a \( 3 \, \text{kg} \) mass as in Fig. 13.14. What is the frequency?

Q7. A particle executes SHM with an amplitude of \( 5 \, \text{cm} \) and a period of \( 2 \, \text{s} \). What is the maximum speed of the particle? (Take \( \pi = 3.14 \))

Q8. What underlies the ability of SHM to be represented as a projection of uniform circular motion?

Q9. What is the physical significance of the angular frequency in SHM?

Q10. In SHM, what is the relationship between the magnitudes of maximum acceleration and maximum velocity?

Q11. A spring-mass system has \( m = 0.2 \, \text{kg}, k = 80 \, \text{N/m} \). If displaced by \( 5 \, \text{cm} \), what is the total energy?

Q12. A spring system has \( k = 500 \, \text{N/m}, m = 2 \, \text{kg} \). What is the maximum acceleration if \( A = 8 \, \text{cm} \)?

Q13. A mass oscillates with \( v = -9 \sin (3t) \) (in m/s). What is its displacement function?

Q14. A particle’s displacement is \( x = 6 \cos (\pi t + \frac{\pi}{6}) \) (in m). What is its velocity at \( t = 0.5 \, \text{s} \)? (Take \( \sin 60^\circ = \frac{\sqrt{3}}{2} \))

Q15. A simple pendulum has a period of \( 3 \, \text{s} \) on Earth (\( g = 9.8 \, \text{m/s}^2 \)). What is its length?