Name: Condition for Resonance
Uploaded: 2019-07-19
Duration: 5 min 17 s
Description: Do you know you can shatter the glass goblet without touching it? Yes, it is possible when the sound of a specific frequency is produced. Let's find out the...

Question

Show that total energy of a simple harmonically vibrating body remains constant during its motion.

Accepted Answer

Consider a particle of mass $m$ , executing linear simple harmonic motion of angular frequency ( $ω$ ) and amplitude ( $A$ ) having displacement $(\vec{x})$ , velocity $(\vec{v})$ and acceleration $(\vec{a})$ at any time $t$ .

Kinetic energy = $K . E = \frac{1}{2} m ω^{2} (A^{2} - x^{2})$

Potential energy possessed by the body is equal to the negative of the total work done by the restoring force When the particle has been displaced from $x$ to $x + d x$ , the work done by restoring force is:

$W = - \frac{m ω^{2} x^{2}}{2}$

Total energy possessed by a body is equal to the sum of potential and kinetic energy of the body.

$\Rightarrow E = K . E + P . E$

$E = \frac{1}{2} m ω^{2} (A^{2} - x^{2}) + \frac{1}{2} m ω^{2} x^{2}$

$E = \frac{1}{2} m ω^{2} A^{2}$

For a given S.H.M, the angular frequency and amplitude are constant quantities. The expression on R.H.S of the given equation is, therefore, a constant quantity.

Hence, the total energy of the particle in SHM is constant, and it is independent of the instantaneous displacement. That means the total energy of a simple harmonically vibrating body remains constant during its motion.

Show that total energy of a simple harmonically vibrating body remains constant during its motion.

Important Questions on Oscillations

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