Name: Damped Harmonic Oscillations
Uploaded: 2019-07-19
Duration: 9 min 23 s
Description: An object floating in the water, when pushed slightly, starts oscillation but comes to rest after some time. This is the concept of damped oscillation. Let's...

Question

Derive the equation for the Kinetic energy and potential energy for a simple harmonic oscillator and show that the total energy of a particle in simple harmonic motion is constant at any point on its path.

Accepted Answer

Let a particle with mass $m$ , angular frequency $ω$ be displaced from its mean position along the x-axis, such that its displacement is $x$ .

The particle executes simple harmonic motion.
Acceleration of the particle, $a = - ω ² x$
The restoring force acting on the particle , $F = m a$

$= - m ω ² x$
Since the displacement and restoring force act in the opposite directions, thus, the work done in displacing the particle through a small distance $d x$ will be, $d W = F . d x \cos 180 °$

Substitute the value of force from the above equation, and integrate it to get the Total work done. $∴ W = \int m ω^{2} x . d x$

$= m ω^{2} x^{2} / 2$ ----(1)

The equation of motion of particle, $x = A \sin (ω t + ϕ)$ , where $A$ is the amplitude and $ϕ$ is the phase constant.

Substitute this value in the equation (1), we get:
$W = \frac{m ω ²}{2} {(A \sin (ω t + ϕ))}^{2}$

$= \frac{m ω ²}{2} A^{2} \sin^{2} (ω t + ϕ)$
Work done is stored in the system as its Potential energy, Thus,
$P . E = W = \frac{m ω ²}{2} A^{2} \sin^{2} (ω t + ϕ)$

If $v$ is the velocity of particle undergoing SHM,

The Kinetic energy of the system, $K . E = \frac{1}{2} m v ²$
$v = \frac{d x}{d t}$

$\Rightarrow v = ω A \cos (ω t + ϕ)$

Substitute the value of velocity in the expression of Kinetic energy.

$K . E = \frac{m ω ²}{2} A ² \cos ² (ω t + ϕ)$

The total energy is the sum of kinetic and potential energy of the particle at a given instant,

$T . E = K . E + P . E$
$= \frac{m ω ²}{2} A ² \cos ² (ω t + ϕ) + \frac{m ω ²}{2} A^{2} \sin^{2} (ω t + ϕ)$
$= \frac{m ω ²}{2} A^{2} [\cos^{2} (ω t + ϕ) + \sin^{2} (ω t + ϕ)]$

$= \frac{m ω ²}{2} A^{2}$

$\Rightarrow T . E$ = constant

Derive the equation for the Kinetic energy and potential energy for a simple harmonic oscillator and show that the total energy of a particle in simple harmonic motion is constant at any point on its path.

Important Questions on Oscillations

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