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Physics: Problems and Solutions

Simple harmonic motion

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Simple harmonic motion is a type of oscillatory motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of the displacement (see Hooke's law). This becomes the following differential equation:

\vec{F} = m \vec{a} = m \vec{x}'' = -k\vec{x}

which results in the following solution:

 x(t) = A\cos\left(\omega t - \varphi\right),

where A is the amplitude, ω is the angular frequency, equal to 2πf, and φ is the phase. ω is equal to

\omega = \sqrt{ \tfrac{k}{m} }

From this equation, velocity and acceleration can easily be found.

The total energy of the system at any time is

E_{tot} = \tfrac{1}{2} k A^2

Derivation of formulaEdit

Hooke's law can be rewritten as a second-order differential equation.

F = - kx = m \frac{d^2 x}{dt^2}
\frac{d^2 x}{dt^2} + \frac{k}{m} x = 0

This equation will have the characteristic equation

r^2 + \frac{k}{m} = 0
r = \pm \sqrt{-\tfrac{k}{m}} = \pm \sqrt{\tfrac{k}{m}} i = \pm \omega i

The solution to the differential equation is

x(t) = C_1 e^{r_1 t} + C_2 e^{r_2 t} =  C_1 e^{\omega i t} + C_2 e^{-\omega i t}

By using Euler's formula this can be written in the form

x(t) = C_1 (\cos (\omega t) + i \sin(\omega t)) + C_2 (\cos (\omega t) - i \sin(\omega t))
x(t) = (C_1 + C_2) \cos (\omega t) + (C_1 - C_2) \sin (\omega t)
x(t) = C_3 \cos (\omega t) + C_4 \sin (\omega t)

which, by using trigonometric identities, can be written as

x(t) = A \cos (\omega t - \varphi)

where A = \sqrt{C_3^{ \ 2} + C_4^{ \ 2}} and \tan \varphi = \tfrac{C_4}{C_3}.

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