A block whose mass is 1 kg is fastened to a spring. The spring has a spring constant of 100 N m^-1. The block is pulled to a distance x = 2 cm from its equilibrium position at x = 0 on a frictionless surface from rest at t = 0. Calculate the kinetic, potential and total energies of the block when it is 1 cm away from the mean position.

The block executes SHM, its angular frequency is:

ω = √^k/_m = √^{100 N m-1}/_{1 kg} = 10 rad s^-1

Its displacement at any time t is:

x(t) = 0.02 cos(10t)

Therefore, when the particle is 1 cm away from the mean position:

0.01 = 0.02 cos(10t) ⇒ cos(10t) = 0.5 ⇒ sin(10t) = 0.866

Velocity of the block at x = 1 cm:

v = 0.02 × 10 × 0.866 m s^-1 = 0.1732 m s^-1

K.E. of the block:

= ¹/₂ m v² = ¹/₂ × 1 kg × (0.1732)² = 0.0149 J

P.E. of the block:

U = ¹/₂ k x² = ¹/₂ (100 N m^-1 × 0.01 m × 0.01 m) = 0.005 J

Total energy of the block at x = 1 cm:

E = K.E. + P.E. = 0.0199 J