Class 11 > Unit # 11:Oscillations > Energy Conservation in S.H.M.

Talha's Physics Academy

Interconversion of Energy & Total Energy in Simple Harmonic Motion

Video Lecture: Energy in Simple Harmonic Motion

Watch the complete step-by-step video lecture explaining kinetic energy, potential energy, and the conservation of total energy in SHM:

Energy of a Body Executing Simple Harmonic Motion

Let us consider a mass '$m$' connected to one end of a spring whose other end is fixed to a rigid wall, executing Simple Harmonic Motion (SHM) on a frictionless horizontal surface.

I. Kinetic Energy (K.E.)

The instantaneous velocity '$v$' of a body executing SHM at a displacement '$x$' is given by:

$v = \omega \sqrt{x_0^2 - x^2}$

For a spring-mass system, the angular frequency $\omega$ is related by $\omega^2 = \frac{K}{m}$, which gives $m\omega^2 = K$ (where $K$ is the spring constant).

The kinetic energy is defined as:

$\text{K.E.} = \frac{1}{2}mv^2$

Substituting the velocity expression into the kinetic energy formula:

$\text{K.E.} = \frac{1}{2}m \left[\omega \sqrt{x_0^2 - x^2}\right]^2$

$\text{K.E.} = \frac{1}{2}m \omega^2 (x_0^2 - x^2)$

$\text{K.E.} = \frac{1}{2}K(x_0^2 - x^2) \quad \text{--- (i)}$

II. Potential Energy (P.E.)

According to Hooke's Law, the restoring force is given by $F = Kx$.

• At the mean position ($x = 0$): Force $F = 0$
• At the extreme position ($x = x_0$): Force $F = Kx_0$

Therefore, the average force acting on mass '$m$' during displacement '$x$' from the mean position is:

$F_{\text{avg}} = \frac{0 + Kx}{2} = \frac{1}{2}Kx$

Potential energy is defined as the work done against this restoring force:

$\text{P.E.} = \text{Average Force} \times \text{Displacement}$

$\text{P.E.} = \left(\frac{1}{2}Kx\right) \times x$

$\text{P.E.} = \frac{1}{2}Kx^2 \quad \text{--- (ii)}$

III. Total Energy (E)

The total mechanical energy '$E$' of a body executing simple harmonic motion is the sum of its kinetic energy and potential energy at any displacement '$x$' from the mean position:

$E = \text{K.E.} + \text{P.E.}$

Substituting equations (i) and (ii) into the total energy equation:

$E = \frac{1}{2}K(x_0^2 - x^2) + \frac{1}{2}Kx^2$

$E = \frac{1}{2}Kx_0^2 - \frac{1}{2}Kx^2 + \frac{1}{2}Kx^2$

$E = \frac{1}{2}Kx_0^2$

Conclusion: Since the spring constant '$K$' and amplitude '$x_0$' are constant for a given system, the total energy '$E$' remains constant throughout the motion. This proves that energy is conserved in SHM, and the total energy is directly proportional to the square of the amplitude of vibration ($\text{E} \propto x_0^2$).

Fig: Interconversion of Kinetic and Potential Energy during Simple Harmonic Motion.