NEET Oscillations — Set 10

Question 1

A particle’s displacement is $ x = 4 \cos (3\pi t + \frac{\pi}{6}) $ (in m). What is its velocity at $ t = 0 \, \text{s} $? (Take $ \sin 30^\circ = 0.5 $)

Accepted Answer

Velocity: $ v = -\omega A \sin (\omega t + \phi) $. $ A = 4 \, \text{m}, \omega = 3\pi \, \text{s}^{-1}, \phi = \frac{\pi}{6} $. At $ t = 0 $: $ v = -3\pi \times 4 \sin \frac{\pi}{6} = -12\pi \times 0.5 \approx -18.84 \, \text{m/s} $.

Question 2

A particle in SHM has $ x = 4 \cos (3t) $ (in m). What is its maximum acceleration?

Accepted Answer

Maximum acceleration: $ a_{\text{max}} = \omega^2 A $. $ A = 4 \, \text{m}, \omega = 3 \, \text{s}^{-1} $. $ a_{\text{max}} = 3^2 \times 4 = 9 \times 4 = 36 \, \text{m/s}^2 $.

Question 3

In an ideal SHM system, what occurs to the total mechanical energy as the particle moves from the mean position to an extreme position?

Accepted Answer

Total mechanical energy in ideal SHM (no friction) is conserved, remaining constant as kinetic energy converts to potential energy during the motion.

NEET Physics: Oscillations — Practice Set 10

Q1. A particle’s displacement is \( x = 4 \cos (3\pi t + \frac{\pi}{6}) \) (in m). What is its velocity at \( t = 0 \, \text{s} \)? (Take \( \sin 30^\circ = 0.5 \))

Q2. A particle in SHM has \( x = 4 \cos (3t) \) (in m). What is its maximum acceleration?

Q3. In an ideal SHM system, what occurs to the total mechanical energy as the particle moves from the mean position to an extreme position?

Q4. Which of the following best explains why the period of a simple pendulum is independent of its mass?

Q5. The period of a simple pendulum is \( 2 \, \text{s} \) when \( g = 9.8 \, \text{m/s}^2 \). What should be the length of the pendulum?

Q6. A spring-mass system has \( m = 0.4 \, \text{kg}, k = 160 \, \text{N/m} \). If displaced by \( 6 \, \text{cm} \), what is the total energy?

Q7. What distinguishes the energy transformation in SHM from that in uniform circular motion?

Q8. A pendulum has \( L = 2.25 \, \text{m}, g = 9.8 \, \text{m/s}^2 \). What is its angular frequency?

Q9. In SHM, if the particle is at an extreme position, what can be inferred about its kinetic energy?

Q10. A spring-mass system has \( m = 0.9 \, \text{kg}, k = 360 \, \text{N/m} \). If displaced by \( 4 \, \text{cm} \), what is the total energy?

Q11. A spring-mass system oscillates with \( T = 0.7 \, \text{s} \) when \( m = 0.7 \, \text{kg} \). What is the spring constant?

Q12. For a simple pendulum, why does the approximation of SHM break down at large amplitudes?

Q13. What is a key reason the simple pendulum deviates from SHM at large angular displacements?

Q14. A spring-mass system has \( m = 4 \, \text{kg}, k = 400 \, \text{N/m} \). If displaced by \( 20 \, \text{cm} \), what is the potential energy at \( x = 10 \, \text{cm} \)?

Q15. A particle in SHM has \( x = 3 \cos (5t) \) (in m). What is its kinetic energy at \( x = 1.5 \, \text{m} \) if \( m = 2 \, \text{kg} \)?