NEET Oscillations — Set 16

Question 1

A spring-mass system oscillates with $ T = 0.6 \, \text{s} $ when $ m = 0.9 \, \text{kg} $. What is the spring constant?

Accepted Answer

$ T = 2\pi \sqrt{\frac{m}{k}} $. $ 0.6 = 2\pi \sqrt{\frac{0.9}{k}} \Rightarrow \frac{0.6}{2\pi} = \sqrt{\frac{0.9}{k}} $. $ (0.0955)^2 = \frac{0.9}{k} \Rightarrow k = \frac{0.9}{0.00912} \approx 98.68 \, \text{N/m} $.

Question 2

A particle’s x-projection from circular motion is $ x = 5 \cos (4t) $ (in m). What is its maximum speed?

Accepted Answer

Maximum speed: $ v_{\text{max}} = \omega A $. $ A = 5 \, \text{m}, \omega = 4 \, \text{s}^{-1} $. $ v_{\text{max}} = 4 \times 5 = 20 \, \text{m/s} $.

Question 3

A particle’s displacement is $ x = 3 \cos (2\pi t - \frac{\pi}{4}) $ (in m). What is its velocity at $ t = 0.25 \, \text{s} $? (Take $ \sin 45^\circ = \frac{\sqrt{2}}{2} $)

Accepted Answer

Velocity: $ v = -\omega A \sin (\omega t + \phi) $. $ A = 3 \, \text{m}, \omega = 2\pi \, \text{s}^{-1}, \phi = -\frac{\pi}{4} $. At $ t = 0.25 $: $ 2\pi \times 0.25 - \frac{\pi}{4} = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4} $. $ v = -2\pi \times 3 \sin \frac{\pi}{4} = -6\pi \times \frac{\sqrt{2}}{2} \approx -13.32 \, \text{m/s} $.

NEET Physics: Oscillations — Practice Set 16

Q1. A spring-mass system oscillates with \( T = 0.6 \, \text{s} \) when \( m = 0.9 \, \text{kg} \). What is the spring constant?

Q2. A particle’s x-projection from circular motion is \( x = 5 \cos (4t) \) (in m). What is its maximum speed?

Q3. A particle’s displacement is \( x = 3 \cos (2\pi t - \frac{\pi}{4}) \) (in m). What is its velocity at \( t = 0.25 \, \text{s} \)? (Take \( \sin 45^\circ = \frac{\sqrt{2}}{2} \))

Q4. A particle’s displacement in SHM is \( x = 6 \sin (3t) \) (in cm). What is the maximum acceleration?

Q5. Which feature of SHM explains why two particles with identical amplitude and frequency may not reach their extreme positions simultaneously?

Q6. For a pendulum executing small oscillations, what approximation allows its motion to be treated as simple harmonic?

Q7. A mass oscillates with \( T = 0.4 \, \text{s} \) when attached to a spring of \( k = 100 \, \text{N/m} \). What is the mass?

Q8. A pendulum has \( L = 0.5 \, \text{m}, g = 9.8 \, \text{m/s}^2 \). What is its angular frequency?

Q9. In SHM, what is true about the particle’s acceleration when its kinetic energy is at its maximum?

Q10. A particle in SHM has \( a = -16 x \) (in SI units). What is its frequency?