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

How does the energy of a simple pendulum vary as it moves from one extreme position to the other during its oscillations?

Accepted Answer

The Total energy associated with a particle executing simple harmonic motion, at any point is the sum of kinetic energy and potential energy at that point.

$T . E = K . E . + P . E$ -----(1)

If the equation of motion of particle is given by $y = A \sin (ω t + ϕ)$

where $A$ is the amplitude and $ϕ$ is the phase constant.

Then, Kinetic energy of the particle = $K . E = \frac{1}{2} m ω^{2} (A^{2} - y^{2})$ ------(2)

Potential energy of the particle = $P . E = \frac{1}{2} m ω^{2} y^{2}$ --------(3)

Substitute these values in equation (1)

So, total energy, $T . E = \frac{1}{2} m ω^{2} (A^{2} - y^{2}) + \frac{1}{2} m ω^{2} y^{2}$

$= \frac{1}{2} m ω^{2} A^{2}$

At mean position, $y = 0$
Therefore, from equation (3),

$P . E = 0$
$\Rightarrow K . E = \frac{1}{2} m ω^{2} A^{2}$

At extreme position, $y = A$

Therefore, from equation (2),

$K . E = 0$
$\Rightarrow P . E = \frac{1}{2} m ω^{2} A^{2}$

This shows that as particle goes from mean position to extreme position, kinetic energy is to be converted into potential energy and vice versa.

How does the energy of a simple pendulum vary as it moves from one extreme position to the other during its oscillations?

Important Questions on Oscillations

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