Simple Harmonic Motion
11.0 Spring Block System
11.0 Spring Block System
Suppose a block of mass ‘$m$’ is attached to the free end of a massless spring of spring constant $k$, with its other end fixed to a rigid support.
Using Force Method:
If the mass is displaced through a distance $x$, as shown, the linear restoring force is, $$F=-kx\ \ \ ..(i)$$
Starts acting on the mass, tending to bring it back into its original position. The negative sign simply indicates that force is acting in the opposite direction to that of the displacement.
Eq. $(i)$ can be written as,
$$\begin{equation} \begin{aligned} ma = - kx \\ a = - \frac{k}{m}x\quad \ldots \left( {ii} \right) \\ a = - {\omega ^2}x\quad ...\left( {iii} \right) \\\end{aligned} \end{equation} $$
On equating $(ii)$ & $(iii)$ we get,
$$\begin{equation} \begin{aligned} \omega = \sqrt {\frac{k}{m}} \quad ...\left( {iv} \right) \\ T = \frac{{2\pi }}{\omega } = 2\pi \sqrt {\frac{m}{k}} \left( {{\text{from eq}}{\text{.}}{\text{ }}iv} \right) \\\end{aligned} \end{equation} $$
Using Energy Method:
The time period of the spring-block system can also be obtained by the energy method.
Let $v$ be the speed of the mass in displaced position. Then total mechanical energy of the spring-block system is,
$$E = {\text{kinetic energy of the block + elastic potential energy of spring}}$$$$E = \frac{1}{2}m{v^2} + \frac{1}{2}k{x^2}$$ Since, $E$ is constant.
$$\begin{equation} \begin{aligned} \frac{{dE}}{{dt}} = 0 \\ \frac{{dE}}{{dt}} = 2.\frac{1}{2}mv\frac{{dv}}{{dt}} + 2.\frac{1}{2}kx\frac{{dx}}{{dt}} \\ 0 = mva + kxv \\ 0 = v\left( {ma + kx} \right) \\ v \ne 0,\;\left( {{\text{as the particle is not at rest}}} \right) \\ \left( {ma + kx} \right) = 0 \\ a = - \frac{k}{m}x\quad ..\left( i \right) \\ a = - {\omega ^2}x\quad ..\left( {ii} \right) \\\end{aligned} \end{equation} $$
From, equation $(i)$ & $(ii)$, we get,
$$\begin{equation} \begin{aligned} \omega = \sqrt {\frac{k}{m}} \quad ..\left( {iii} \right) \\ T = \frac{{2\pi }}{\omega } = 2\pi \sqrt {\frac{m}{k}} \\\end{aligned} \end{equation} $$
