NEET Oscillations — Set 5

Question 1

Why does the period of a spring-mass system differ fundamentally from that of a simple pendulum in terms of gravitational influence?

Accepted Answer

The spring-mass period ($ T = 2\pi \sqrt{\frac{m}{k}} $) lacks gravitational dependence, relying on elasticity, while the pendulum’s period ($ T = 2\pi \sqrt{\frac{L}{g}} $) varies with gravity.

Question 2

A simple pendulum has a frequency of $ 0.3 \, \text{Hz} $ on Earth ($ g = 9.8 \, \text{m/s}^2 $). What is its length?

Accepted Answer

Period: $ T = \frac{1}{v} = \frac{1}{0.3} \approx 3.333 \, \text{s} $. $ T = 2\pi \sqrt{\frac{L}{g}} \Rightarrow 3.333 = 2\pi \sqrt{\frac{L}{9.8}} $. $ \sqrt{\frac{L}{9.8}} = \frac{3.333}{2\pi} \approx 0.531 \Rightarrow \frac{L}{9.8} = (0.531)^2 \Rightarrow L \approx 2.76 \, \text{m} $.

Question 3

Which of the following functions represents SHM? (Assume $ \omega $ is a positive constant)

Accepted Answer

For SHM, acceleration $ a = -\omega^2 x $. Check by differentiating twice: (a) $ x = \sin \omega t + \cos 2\omega t $: Not SHM (different frequencies). (b) $ x = 2 \sin (\omega t + \frac{\pi}{4}) $: $ v = 2\omega \cos (\omega t + \frac{\pi}{4}), a = -2\omega^2 \sin (\omega t + \frac{\pi}{4}) = -\omega^2 x $. SHM. (c) $ x = e^{-\omega t} $: Not periodic, not SHM. (d) $ x = \log (\omega t) $: Not periodic, not SHM.

NEET Physics: Oscillations — Practice Set 5

Q1. Why does the period of a spring-mass system differ fundamentally from that of a simple pendulum in terms of gravitational influence?

Q2. A simple pendulum has a frequency of \( 0.3 \, \text{Hz} \) on Earth (\( g = 9.8 \, \text{m/s}^2 \)). What is its length?

Q3. Which of the following functions represents SHM? (Assume \( \omega \) is a positive constant)

Q4. A spring-mass system has \( m = 1.5 \, \text{kg}, k = 600 \, \text{N/m} \). What is its period?

Q5. Two springs, each of \( k = 50 \, \text{N/m} \), are connected in parallel to a \( 2 \, \text{kg} \) mass. What is the period of oscillation?

Q6. In a simple pendulum executing SHM, what ensures that the period is independent of the initial angular displacement?

Q7. A particle in SHM has \( x = 3 \cos (2t + \frac{\pi}{6}) \) (in m). What is its speed at \( t = 0.5 \, \text{s} \)? (Take \( \sin \frac{4\pi}{6} = 1 \))

Q8. A spring-mass system has \( T = 0.2 \, \text{s}, m = 0.8 \, \text{kg} \). What is the spring constant?

Q9. A particle in SHM has an amplitude of \( 12 \, \text{cm} \) and a period of \( 1.2 \, \text{s} \). What is its maximum velocity? (Take \( \pi = 3.14 \))

Q10. In SHM, why does the particle’s velocity lead its displacement by \( \pi/2 \) radians?

Q11. A mass of \( 2 \, \text{kg} \) is attached to a spring with \( k = 200 \, \text{N/m} \). What is the frequency of oscillation?

Q12. A particle in SHM has \( x = 6 \sin (2\pi t + \frac{\pi}{4}) \) (in cm). What is its acceleration at \( t = 0.25 \, \text{s} \)?

Q13. A particle’s displacement is \( x = 7 \sin (2t + \frac{\pi}{4}) \) (in m). What is its velocity at \( t = 0 \, \text{s} \)? (Take \( \cos 45^\circ = \frac{\sqrt{2}}{2} \))

Q14. A particle in SHM has \( x = 3 \sin (4t + \frac{\pi}{3}) \) (in m). What is its acceleration at \( t = 0.25 \, \text{s} \)? (Take \( \cos 60^\circ = 0.5 \))

Q15. Why does the frequency of a spring-mass system increase when a stiffer spring is used?