17.7 Energy Conservation in SHM | Simple Harmonic Motion | Physics 12

In an ideal (frictionless) simple harmonic oscillator, the total mechanical energy is conserved. Energy continuously interchanges between kinetic energy (K.E.) and potential energy (P.E.), but their sum remains constant at all times.

Potential Energy in SHM

For a mass-spring system, the elastic potential energy stored when the mass is at displacement $x$ from the mean position is:

$P . E . = \frac{1}{2} k x^{2}$

Key observations:

At the mean position ( $x = 0$ ): $P . E . = 0$ (minimum)
At the extreme positions ( $x = \pm x_{0}$ ): $P . E . = \frac{1}{2} k x_{0}^{2}$ (maximum)

The P.E. varies as the square of displacement, so its graph against $x$ is a parabola opening upwards.

Kinetic Energy in SHM

The instantaneous velocity at displacement $x$ is:

$v = ω x_{0}^{2} - x^{2}$

Substituting into $K . E . = \frac{1}{2} m v^{2}$ and using $k = m ω^{2}$ :

$K . E . = \frac{1}{2} m ω^{2} (x_{0}^{2} - x^{2}) = \frac{1}{2} k (x_{0}^{2} - x^{2})$

Key observations:

At the mean position ( $x = 0$ ): $K . E . = \frac{1}{2} k x_{0}^{2}$ (maximum)
At the extreme positions ( $x = \pm x_{0}$ ): $K . E . = 0$ (minimum)

The K.E. graph against $x$ is a downward-opening parabola.

Total Mechanical Energy

Adding P.E. and K.E. at any displacement $x$ :

$E_{t o t a l} = K . E . + P . E . = \frac{1}{2} k (x_{0}^{2} - x^{2}) + \frac{1}{2} k x^{2}$

$E_{t o t a l} = \frac{1}{2} k x_{0}^{2} = \frac{1}{2} m ω^{2} x_{0}^{2}$

This result is independent of displacement — the total energy depends only on the spring constant $k$ , the mass $m$ , the angular frequency $ω$ , and the amplitude $x_{0}$ .

Important: Since $E_{t o t a l} \propto x_{0}^{2}$ , doubling the amplitude quadruples the total energy.

Energy Interchange During SHM

Position	K.E.	P.E.	Total E
Mean position ( $x = 0$ )	Maximum $= \frac{1}{2} k x_{0}^{2}$	Zero	$\frac{1}{2} k x_{0}^{2}$
Extreme position ( $x = \pm x_{0}$ )	Zero	Maximum $= \frac{1}{2} k x_{0}^{2}$	$\frac{1}{2} k x_{0}^{2}$
$x = x_{0} / 2$	$\frac{1}{4} k x_{0}^{2}$	$\frac{1}{4} k x_{0}^{2}$	$\frac{1}{2} k x_{0}^{2}$

At $x = x_{0} / 2$ , the kinetic and potential energies are equal, each equal to half the total energy.

Conservation of Energy (No Dissipation)

In the absence of friction or air resistance, no energy is lost. The total mechanical energy remains constant:

$E_{t o t a l} = K . E . + P . E . = constant = \frac{1}{2} k x_{0}^{2}$

In real systems, resistive forces cause damping, gradually reducing the amplitude and total energy over time.

Potential Energy in SHM

For a mass-spring system, the elastic potential energy stored when the mass is at displacement $x$ from the mean position is:

$P . E . = \frac{1}{2} k x^{2}$

Key observations:

At the mean position ( $x = 0$ ): $P . E . = 0$ (minimum)
At the extreme positions ( $x = \pm x_{0}$ ): $P . E . = \frac{1}{2} k x_{0}^{2}$ (maximum)

The P.E. varies as the square of displacement, so its graph against $x$ is a parabola opening upwards.

Kinetic Energy in SHM

The instantaneous velocity at displacement $x$ is:

$v = ω x_{0}^{2} - x^{2}$

Substituting into $K . E . = \frac{1}{2} m v^{2}$ and using $k = m ω^{2}$ :

$K . E . = \frac{1}{2} m ω^{2} (x_{0}^{2} - x^{2}) = \frac{1}{2} k (x_{0}^{2} - x^{2})$

Key observations:

At the mean position ( $x = 0$ ): $K . E . = \frac{1}{2} k x_{0}^{2}$ (maximum)
At the extreme positions ( $x = \pm x_{0}$ ): $K . E . = 0$ (minimum)

The K.E. graph against $x$ is a downward-opening parabola.

Total Mechanical Energy

Adding P.E. and K.E. at any displacement $x$ :

$E_{t o t a l} = K . E . + P . E . = \frac{1}{2} k (x_{0}^{2} - x^{2}) + \frac{1}{2} k x^{2}$

$E_{t o t a l} = \frac{1}{2} k x_{0}^{2} = \frac{1}{2} m ω^{2} x_{0}^{2}$

This result is independent of displacement — the total energy depends only on the spring constant $k$ , the mass $m$ , the angular frequency $ω$ , and the amplitude $x_{0}$ .

Important: Since $E_{t o t a l} \propto x_{0}^{2}$ , doubling the amplitude quadruples the total energy.

Energy Interchange During SHM

Position	K.E.	P.E.	Total E
Mean position ( $x = 0$ )	Maximum $= \frac{1}{2} k x_{0}^{2}$	Zero	$\frac{1}{2} k x_{0}^{2}$
Extreme position ( $x = \pm x_{0}$ )	Zero	Maximum $= \frac{1}{2} k x_{0}^{2}$	$\frac{1}{2} k x_{0}^{2}$
$x = x_{0} / 2$	$\frac{1}{4} k x_{0}^{2}$	$\frac{1}{4} k x_{0}^{2}$	$\frac{1}{2} k x_{0}^{2}$

At $x = x_{0} / 2$ , the kinetic and potential energies are equal, each equal to half the total energy.

Conservation of Energy (No Dissipation)

In the absence of friction or air resistance, no energy is lost. The total mechanical energy remains constant:

$E_{t o t a l} = K . E . + P . E . = constant = \frac{1}{2} k x_{0}^{2}$

In real systems, resistive forces cause damping, gradually reducing the amplitude and total energy over time.