Amass is attached to an ideal spring. at time t = 0 the spring is at its natural length and the mass is given an initial velocity; the period of the ensuing (one-dimensional) simple harmonic motion is t . at what time is the power delivered to the mass by the spring first a maximum?

t = T/4

Explanation:

The power delivered to the mass by the spring is work done by the spring per second.

The work done by the spring is equal to the elastic potential energy stored in the spring.

The maximum energy stored in the spring is at the amplitude of the oscillation.

So the first time the mass reaches to its amplitude can be found by the following equation of motion:

When the mass reaches the amplitude:

because cos(π) = 1.

Using ω = 2π/T,

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