Simple Harmonic Motion
8.0 Force and Energy in Simple Harmonic Motion
8.0 Force and Energy in Simple Harmonic Motion
We know that the acceleration of the SHM is $$a = - {\omega ^2}x$$ By applying equation of motion, $$\vec F = m\vec a$$we have $$\vec F = - m{\omega ^2}x = - kx$$ After solving we get, $$\omega = \sqrt {\frac{k}{m}} $$
Thus in SHM the force is proportional and opposite to the displacement.
So, when the displacement is to the right of the mean position, the force will act along the left direction and vice versa.
Thus the force always point towards the origin $O$. Such type of force appears when an elastic body such as spring is deformed.
That is why the constant, $k = m{\omega ^2}$ is sometimes called the elastic constant.
Further,
$$\begin{equation} \begin{aligned} \omega = \frac{{2\pi }}{T} = \sqrt {\frac{k}{m}} \\ T = 2\pi \sqrt {\frac{m}{k}}\ and,\ f = \frac{1}{{2\pi }}\sqrt {\frac{k}{m}} \\\end{aligned} \end{equation} $$
