High School Physics/Simple Oscillation

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Simple Oscillation[edit]

For a simple oscillator consisting of a mass m to one end of a spring with a spring constant s, the restoring force, f, can be expressed by the equation

Force = Spring constant x displacement\,

where displacement is the displacement of the mass from its rest position. Substituting the expression for f into the linear momentum equation,

f = ma = {d^2x \over dt^2}\,

where a is the acceleration of the mass, we can get

m\frac{d^2 x}{d t^2 }= -sx


\frac{d^2 x}{d t^2} + \frac{s}{m}x = 0

Note that

\omega_0^2 = {s \over m}\,

To solve the equation, we can assume

x(t)=A e^{\lambda t} \,

The general solution for this type of 'simple harmonic motion' is x=Asin(wt+phi). Here, phi(the angle expressed in radians) is known as the phase of the simple harmonic motion.