Harmonic motion and circular motion

Let us project an object in uniform circular motion on a straight line (eg, watch its shadow on the wall). The angular position of the object is the initial angular position plus the angular displacement since then (in time \(t\)):

\[\phi(t) = \phi_0 + \Delta \phi = \phi_0 + \omega t,\]

where \(\omega\) is the angular frequency of the uniform circular motion.

The \(x\) component of the acceleration is

\[a_x = -a \cdot \sin\phi = -\omega^2 \cdot r \cdot \sin\phi.\]

The \(x\) coordinate of the position is a side of a right-angled triangle with hypotenuse \(r\), opposite the angle \(\phi\):

\[\sin\phi = \frac{x(t)}{r}.\]

The relationship between the \(x\) component of the acceleration and the \(x\) coordinate:

\[a_x(t) = -\omega^2 \cdot r \cdot \sin\phi = -\omega^2 \cdot r \cdot \frac{x(t)}{r} = -\omega^2 \cdot x(t).\]

Which means that the projection of uniform circular motion on a straight axis is simple harmonic motion.

The maximum value of the \(x\) coordinate is the radius of the circular motion:

\[A = x_{\mathrm{max}} = r.\]

The \(x\) coordinate at time \(t\) is

\[x(t) = r \sin\phi = A \cdot \sin\left(\phi_0 + \omega t\right).\]

The \(x\) component of the velocity at time \(t\):

\[v_x(t) = v \cos\phi = \omega \cdot r \cdot \cos\left(\phi_0 + \omega t\right) = A \cdot \omega \cdot \cos\left(\phi_0 + \omega t\right).\]

The \(x\) component of the acceleration at time \(t\):

\[a_x(t) = -\omega^2 \cdot r \cdot \sin\phi = -A \cdot \omega^2 \sin\left(\phi_0 + \omega t\right) = -\omega^2 \cdot x(t).\]