A particle executes simple harmonic motion of time period T . At t = 0 , it starts from the extreme position. If the maximum kinetic energy of the particle is K₀ , the work done by the restoring force during the time interval t = 0 to t = T 6 is

Step-by-step solution

The equation of motion for a particle starting from the extreme position is x(t) = A ( 2 T t ) . At t = 0 , the position is x = A . The initial potential energy is U_i = 1 2 m ^2 A^2 = K₀ (since maximum kinetic energy equals total energy). At t = T 6 , the position is: x = A ( 2 T T 6 ) = A ( 3 ) = A 2 The final potential energy at this instant is: U_f = 1 2 m ^2 ( A 2 )^2 = 1 4 ( 1 2 m ^2 A^2 ) = K₀ 4 The work done by the conservative restoring force is equal to the negative of the change in potential energy: W =