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PhysicsClassical Physics

What Is Simple Harmonic Motion? Derive x(t), v(t), and a(t)

The oscillator has angular frequency 4.00 rad/s, with x(t) = 0.0200 cos(4.00t) m, v(t) = −0.0800 sin(4.00t) m/s, and a(t) = −0.320 cos(4.00t) m/s².

Question

A 2.00 kg2.00\,\mathrm{kg} block rests on a frictionless horizontal surface and is attached to a spring with force constant k=32.00 N/mk=32.00\,\mathrm{N/m}. The equilibrium position is x=0x=0. The block is pulled to x=+0.020 mx=+0.020\,\mathrm{m} and released from rest at t=0t=0. It oscillates between +0.020 m+0.020\,\mathrm{m} and −0.020 m-0.020\,\mathrm{m} with period T=1.57 sT=1.57\,\mathrm{s}.

Determine the position, velocity, and acceleration as functions of time.

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Evidence boundary

The equations describe an ideal undamped horizontal mass-spring oscillator obeying Hooke’s law, with t = 0 at maximum positive displacement. They use the stated period, which is consistent with sqrt(k/m) to the given precision. Damping, driving, spring mass, friction, and nonlinear motion are excluded.