The displacement of a particle in SHM is x = 10sin\(\left(2 t-\frac{\pi}{6}\right)\) metre. When its displacement is 6 m, the velocity of the particle (in m s-1) is

Calculation:
The displacement of a particle in Simple Harmonic Motion (SHM) is given by the equation:

x = 10sin(2t - π/6) metres

To find the velocity, we differentiate the displacement equation with respect to time.

The velocity (v) is given by:

v = dx/dt = d/dt [10sin(2t - π/6)]

Using the chain rule:

v = 10 × 2cos(2t - π/6)

v = 20cos(2t - π/6)

When the displacement is 6 m, we substitute this into the displacement equation:

6 = 10sin(2t - π/6)

sin(2t - π/6) = 6/10 = 0.6

2t - π/6 = sin⁻¹(0.6)

2t - π/6 = 0.6435 rad

2t = 0.6435 + π/6

2t = 0.6435 + 0.5236 = 1.1671 rad

t = 1.1671 / 2 = 0.5836 s

Now, substitute t = 0.5836 s into the velocity equation:

v = 20cos(2 × 0.5836 - π/6)

v = 20cos(1.1671 - 0.5236)

v = 20cos(0.6435) = 20 × 0.8 = 16 m/s

The velocity of the particle when its displacement is 6 m is 16 m/s.