NEET Oscillations — Set 12

Question 1

A particle in SHM has $ x = 7 \sin (3t + \frac{\pi}{6}) $ (in m). What is its acceleration at $ t = 0 \, \text{s} $? (Take $ \sin 30^\circ = 0.5 $)

Accepted Answer

Acceleration: $ a = -\omega^2 x $. $ \omega = 3 \, \text{s}^{-1}, x(0) = 7 \sin \frac{\pi}{6} = 7 \times 0.5 = 3.5 \, \text{m} $. $ a = -3^2 \times 3.5 = -9 \times 3.5 = -31.5 \, \text{m/s}^2 $.

Question 2

A spring-mass system has $ m = 0.6 \, \text{kg}, k = 240 \, \text{N/m} $. If displaced by $ 7 \, \text{cm} $, what is the total energy?

Accepted Answer

Total energy: $ E = \frac{1}{2} k A^2 $. $ A = 0.07 \, \text{m}, k = 240 \, \text{N/m} $. $ E = 0.5 \times 240 \times (0.07)^2 = 0.5 \times 240 \times 0.0049 = 0.588 \, \text{J} $.

Question 3

A particle’s motion is given by $ x = 2 \cos (5t - \frac{\pi}{4}) $ (in m). What is its kinetic energy at $ x = 1 \, \text{m} $ if $ m = 1 \, \text{kg} $?

Accepted Answer

Total energy: $ E = \frac{1}{2} k A^2 = \frac{1}{2} m \omega^2 A^2 = 0.5 \times 1 \times 5^2 \times 2^2 = 50 \, \text{J} $. Potential energy: $ U = \frac{1}{2} m \omega^2 x^2 = 0.5 \times 1 \times 25 \times 1^2 = 12.5 \, \text{J} $. Kinetic energy: $ K = E - U = 50 - 12.5 = 37.5 \, \text{J} $.

NEET Physics: Oscillations — Practice Set 12

Q1. A particle in SHM has \( x = 7 \sin (3t + \frac{\pi}{6}) \) (in m). What is its acceleration at \( t = 0 \, \text{s} \)? (Take \( \sin 30^\circ = 0.5 \))

Q2. A spring-mass system has \( m = 0.6 \, \text{kg}, k = 240 \, \text{N/m} \). If displaced by \( 7 \, \text{cm} \), what is the total energy?

Q3. A particle’s motion is given by \( x = 2 \cos (5t - \frac{\pi}{4}) \) (in m). What is its kinetic energy at \( x = 1 \, \text{m} \) if \( m = 1 \, \text{kg} \)?

Q4. A pendulum oscillates with \( \theta_{\text{max}} = 0.2 \, \text{rad}, L = 2 \, \text{m}, g = 10 \, \text{m/s}^2 \). What is its maximum speed?

Q5. A particle in SHM has \( a = -9 x \) (in SI units). What is its period?

Q6. A particle in SHM has an amplitude of \( 8 \, \text{cm} \) and a frequency of \( 2 \, \text{Hz} \). What is its maximum acceleration? (Take \( \pi = 3.14 \))

Q7. A spring-mass system has \( m = 0.25 \, \text{kg}, k = 100 \, \text{N/m} \). If displaced by \( 6 \, \text{cm} \), what is the total energy?

Q8. A particle’s displacement is \( x = 5 \sin (\pi t + \frac{\pi}{3}) \) (in m). What is its velocity at \( t = 0 \, \text{s} \)? (Take \( \cos \frac{\pi}{3} = 0.5 \))

Q9. A pendulum of length \( 2 \, \text{m} \) oscillates with \( g = 10 \, \text{m/s}^2 \). What is its frequency?

Q10. What effect does tripling the spring constant have on the period of a spring-mass system if the mass is also tripled?