NEET Oscillations — Set 18

Question 1

A particle in SHM has $ x = 3 \cos (2\pi t + \frac{\pi}{3}) $ (in m). What is its speed at $ t = 0.5 \, \text{s} $? (Take $ \sin 120^\circ = \frac{\sqrt{3}}{2} $)

Accepted Answer

Velocity: $ v = -\omega A \sin (\omega t + \phi) $. $ A = 3 \, \text{m}, \omega = 2\pi \, \text{s}^{-1}, \phi = \frac{\pi}{3} $. At $ t = 0.5 $: $ 2\pi \times 0.5 + \frac{\pi}{3} = \pi + \frac{\pi}{3} = \frac{4\pi}{3} $. $ v = -2\pi \times 3 \sin \frac{4\pi}{3} = -6\pi \sin (180^\circ - 60^\circ) = -6\pi \left(-\frac{\sqrt{3}}{2}\right) \approx 16.31 \, \text{m/s} $.

Question 2

A mass of $ 2.0 \, \text{kg} $ on a spring with $ k = 800 \, \text{N/m} $ has $ A = 5 \, \text{cm} $. What is the kinetic energy at $ x = 2.5 \, \text{cm} $?

Accepted Answer

Total energy: $ E = \frac{1}{2} k A^2 = 0.5 \times 800 \times (0.05)^2 = 1 \, \text{J} $. Potential energy: $ U = \frac{1}{2} k x^2 = 0.5 \times 800 \times (0.025)^2 = 0.25 \, \text{J} $. Kinetic energy: $ K = E - U = 1 - 0.25 = 0.75 \, \text{J} $.

Question 3

A pendulum has $ L = 1.96 \, \text{m}, g = 9.8 \, \text{m/s}^2 $. What is its angular frequency?

Accepted Answer

$ \omega = \sqrt{\frac{g}{L}} = \sqrt{\frac{9.8}{1.96}} = \sqrt{5} \approx 2.24 \, \text{rad/s} $.

NEET Physics: Oscillations — Practice Set 18

Q1. A particle in SHM has \( x = 3 \cos (2\pi t + \frac{\pi}{3}) \) (in m). What is its speed at \( t = 0.5 \, \text{s} \)? (Take \( \sin 120^\circ = \frac{\sqrt{3}}{2} \))

Q2. A mass of \( 2.0 \, \text{kg} \) on a spring with \( k = 800 \, \text{N/m} \) has \( A = 5 \, \text{cm} \). What is the kinetic energy at \( x = 2.5 \, \text{cm} \)?

Q3. A pendulum has \( L = 1.96 \, \text{m}, g = 9.8 \, \text{m/s}^2 \). What is its angular frequency?

Q4. A spring-mass system has \( m = 0.25 \, \text{kg} \) and \( k = 100 \, \text{N/m} \). What is its period of oscillation?

Q5. A particle in SHM follows \( x = 2 \sin (4t + \frac{\pi}{2}) \) (in m). What is its acceleration at \( t = 0 \, \text{s} \)?

Q6. A mass of \( 2 \, \text{kg} \) on a spring with \( k = 200 \, \text{N/m} \) has \( A = 10 \, \text{cm} \). What is the kinetic energy at \( x = 5 \, \text{cm} \)?

Q7. A spring system has \( m = 1.0 \, \text{kg}, k = 400 \, \text{N/m}, A = 6 \, \text{cm} \). What is the potential energy at \( x = 3 \, \text{cm} \)?

Q8. A particle’s displacement is \( x = 5 \sin (2\pi t + \frac{\pi}{6}) \) (in m). What is its velocity at \( t = 0.5 \, \text{s} \)? (Take \( \sin 60^\circ = \frac{\sqrt{3}}{2} \))

Q9. A particle in SHM has \( a = -64 x \) (in SI units). What is its period?

Q10. A simple pendulum has a length of \( 2.25 \, \text{m} \) and oscillates with \( g = 9.8 \, \text{m/s}^2 \). What is its period?