The displacement of a particle executing SHM is given by x=10 sin (w t+π/3) m. The time period of motion is 3.14 s. The velocity of the particle at t=0 is _______ m / s.

Solution

Physics sir · 4 steps

Differentiate x once, then put t = 0. The phase π/3 decides how much of the full speed you get.

1. Find omega. The angular frequency is ω = 2π/T = (2 × 3.14)/3.14 = 2 rad s⁻¹.
2. Velocity equation. Velocity is the time derivative of displacement. The derivative of sin is cos, and the chain rule brings out a factor ω.

   v = dx/dt = 10 ω cos(ω t + π/3)

3. Put t = 0. v = 10 × 2 × cos π/3 = 20 × 1/2 = 10 m s⁻¹.
4. So. The velocity of the particle at t = 0 is 10 m s⁻¹.

The common slip: 20. Using the maximum speed Aω and ignoring the phase gives 20, not 10. At t = 0 the particle is not at the mean position, so multiply by cos π/3 = 1/2.

The idea to keep. For x = A sin(ω t + φ), velocity is Aω cos(ω t + φ). Get ω from the period, then put in the time. Never skip the phase.

Answer from the NTA final answer key. Solution by PrepWiser.