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

Calculate the kinetic and potential energies of a simple harmonically vibrating body as a function of time and as a function of displacement of the body.

Accepted Answer

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

$x = A \sin (ω t + ϕ)$

$v = A ω \cos (ω t + ϕ) = ω \sqrt{A^{2} - x^{2}}$

$a = - ω^{2} A \sin (ω t + ϕ) = - ω^{2} x$

Kinetic energy = $\frac{1}{2} m v^{2}$

Substituting the values from above,

$= \frac{1}{2} m {(A ω \cos (ω t + ϕ))}^{2} = \frac{1}{2} m A^{2} ω^{2} \cos^{2} (ω t + ϕ)$

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

Potential energy possessed by the body is equal to the negative of the total work done by the restoring force in displacing the particle from its mean position $(x = 0)$ to extreme position $(x = x)$ :

When the particle has been displaced from $x$ to $x + d x$ , the work done by restoring force is:

$d W = F . d x = - k x d x$

$W = \int d W = \int_{0}^{x} - k x d x = \frac{- k x^{2}}{2}$

$= - \frac{m ω^{2} x^{2}}{2} [k = m ω^{2}]$

$= - \frac{m ω^{2}}{2} A^{2} \sin^{2} (ω t + ϕ)$

$\Rightarrow P . E = - W$

Potential Energy = $\frac{m ω^{2} x^{2}}{2}$

$= \frac{m ω^{2} A^{2}}{2} \sin^{2} (ω t + ϕ)$

Calculate the kinetic and potential energies of a simple harmonically vibrating body as a function of time and as a function of displacement of the body.

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