Question

The angular frequency and the initial phase of the particle performing simple harmonic motion are π rad/sec and \(\frac{\pi}{2}\) rad respectively. The maximum displacement will take place after a time interval of:
1. 1 sec
2. 0.5 sec
3. 1.5 sec
4. 0.25 sec

Answer 1

Correct Answer - Option 2 : 0.5 sec

CONCEPT:

Simple harmonic motion

• Simple Harmonic Motion or SHM is defined as a motion in which the restoring force is directly proportional to the displacement of the body from its mean position.
  • The direction of this restoring force is always towards the mean position.
• Amplitude: It is the maximum displacement of the particle.
• Time period: The minimum time after which the particle keeps on repeating its motion is known as the time period.
• The equation of simple harmonic motion is given as,

⇒ y = a.sin(ωt + ϕ)

Where y = displacement at time t, a = amplitude, ω = angular frequency, and ϕ = initial phase

CALCULATION:

Given ω = π rad/sec, and ϕ = \(\frac{\pi}{2}\) rad

• The equation of simple harmonic motion is given as,

⇒ y = a.sin(ωt + ϕ)     -----(1)

For maximum displacement,

⇒ sin(ωt + ϕ) = 1

⇒ ωt + ϕ = π

⇒ πt + \(\frac{\pi}{2}\) = π

⇒ πt = \(\frac{\pi}{2}\)

\(⇒ t=\frac{1}{2}\,sec\)

⇒ t = 0.5 sec

• Hence, option 2 is correct.