Starting from the origin, a body oscillates simple harmonically with a period of 2s. After what time will its kinetic energy be 75% of the total energy?

In simple harmonic motion, the kinetic energy (KE) and potential energy (PE) vary sinusoidally and are 90 degrees out of phase. The total energy (TE) is constant and is the sum of the kinetic and potential energy.

The kinetic energy is given by KE = TE * (1 - cos^2(wt)), where w is the angular frequency and t is the time.

We are given that KE is 75% of TE, so we can set up the equation 0.75 = 1 - cos^2(wt).

Solving for cos^2(wt) gives us cos^2(wt) = 0.25.

Taking the square root of both sides gives us cos(wt) = ±0.5.

The cosine function equals 0.5 at angles of 60 degrees and 300 degrees, or π/3 and 5π/3 radians.

The period of the motion is given as 2 seconds, so the angular frequency w is 2π/2 = π rad/s.

Setting wt equal to π/3 and 5π/3 and solving for t gives us t = 1/3 s and t = 5/3 s.

Therefore, the kinetic energy is 75% of the total energy at times 1/3 s and 5/3 s after starting from the origin.