Question

Derive the equations to find the position, velocity, and acceleration in simple harmonic motions.

Answer 1

For one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear ordinary differential equation with constant coefficients, can be obtained by means of Newton's second law:

where m is the inertial mass of the oscillating body, x is its displacement from the equilibrium (or mean) position, and k is a constant (the spring constant for a mass on a spring).

Therefore,

Solving the differential equation above produces a solution that is a sinusoidal function.

where

In the solution, c₁ and c₂ are two constants determined by the initial conditions, and the origin is set to be the equilibrium position. Each of these constants carries a physical meaning of the motion: A is the amplitude (maximum displacement from the equilibrium position), ω = 2πf is the angular frequency, and φ is the phase.

Using the techniques of calculus, the velocity and acceleration as a function of time can be found:

Speed:

Maximum speed (at equilibrium point)

Maximum acceleration = (at extreme points).