Simple Harmonic Motion

Question

What is the differential equation that defines simple harmonic oscillation, and its general solution. State two physical conditions that must be satisfied for simple harmonic motion to occur in a mechanical system.

Answer 1

Consider a spring-mass system where the only force present is the restoring force caused by displacing the mass from its equilibrium position.

 

k = stiffness of the spring

m = mass

x = displacement of mass from its equilibrium position

F = force

 

F = -kx

 

By Newton's 2nd law,

 

m d²x/dt² = -kx

 

d²x/dt² + (k/m)x = 0

 

This is the differential equation for simple harmonic motion.

 

General solution:

 

x(t) = x_max sin[√(k/m) t + φ]

 

where x_max and φ may be found from initial conditions.

 

A restoring force is required and energy must be conserved.