NEET Oscillations — Set 9

Question 1

A particle’s x-projection from circular motion is $ x = 7 \cos (3t) $ (in m). What is its maximum acceleration?

Accepted Answer

Maximum acceleration: $ a_{\text{max}} = \omega^2 A $. $ A = 7 \, \text{m}, \omega = 3 \, \text{s}^{-1} $. $ a_{\text{max}} = 3^2 \times 7 = 9 \times 7 = 63 \, \text{m/s}^2 $.

Question 2

A mass oscillates with $ v = -6 \sin (4t) $ (in m/s). What is its amplitude?

Accepted Answer

Velocity: $ v = -\omega A \sin (\omega t) $. Given $ \omega = 4 \, \text{s}^{-1}, v_{\text{max}} = \omega A = 6 $. $ A = \frac{6}{4} = 1.5 \, \text{m} $.

Question 3

A mass of $ 1.5 \, \text{kg} $ on a spring with $ k = 150 \, \text{N/m} $ has $ A = 12 \, \text{cm} $. What is the kinetic energy at $ x = 6 \, \text{cm} $?

Accepted Answer

Total energy: $ E = \frac{1}{2} k A^2 = 0.5 \times 150 \times (0.12)^2 = 1.08 \, \text{J} $. Potential energy: $ U = \frac{1}{2} k x^2 = 0.5 \times 150 \times (0.06)^2 = 0.27 \, \text{J} $. Kinetic energy: $ K = E - U = 1.08 - 0.27 = 0.81 \, \text{J} $.

NEET Physics: Oscillations — Practice Set 9

Q1. A particle’s x-projection from circular motion is \( x = 7 \cos (3t) \) (in m). What is its maximum acceleration?

Q2. A mass oscillates with \( v = -6 \sin (4t) \) (in m/s). What is its amplitude?

Q3. A mass of \( 1.5 \, \text{kg} \) on a spring with \( k = 150 \, \text{N/m} \) has \( A = 12 \, \text{cm} \). What is the kinetic energy at \( x = 6 \, \text{cm} \)?

Q4. In SHM, which quantity is maximized when the particle’s displacement is zero?

Q5. In an oscillatory system, if the displacement and acceleration are always in opposite directions, what type of motion is exhibited?

Q6. Two identical springs (\( k = 90 \, \text{N/m} \)) are attached to a \( 1.8 \, \text{kg} \) mass as in Fig. 13.14. What is the period?

Q7. A particle’s displacement is \( x = 3 \cos (4\pi t + \frac{\pi}{3}) \) (in m). What is its velocity at \( t = 0 \, \text{s} \)? (Take \( \sin 60^\circ = \frac{\sqrt{3}}{2} \))

Q8. Two identical springs (\( k = 75 \, \text{N/m} \)) are attached to a \( 1.5 \, \text{kg} \) mass as in Fig. 13.14. What is the frequency?

Q9. Which aspect of SHM ensures that the time period is independent of the initial conditions?

Q10. A simple pendulum has a frequency of \( 0.4 \, \text{Hz} \) on Earth (\( g = 9.8 \, \text{m/s}^2 \)). What is its length?

Q11. Which factor primarily dictates the frequency of oscillation in a spring-mass system?

Q12. A spring of \( k = 400 \, \text{N/m} \) has a \( 2 \, \text{kg} \) mass. If \( E = 2 \, \text{J} \), what is the amplitude?

Q13. What causes the restoring force in a spring-mass system to initiate SHM?

Q14. A particle in SHM has an amplitude of \( 9 \, \text{cm} \) and a period of \( 0.8 \, \text{s} \). What is its maximum velocity? (Take \( \pi = 3.14 \))

Q15. A mass of \( 1 \, \text{kg} \) on a spring with \( k = 100 \, \text{N/m} \) has \( A = 15 \, \text{cm} \). What is the kinetic energy at \( x = 7.5 \, \text{cm} \)?