Question

When a particle executing SHM oscillates with a frequency $\nu$, then the kinetic energy of the particle:

• A. changes periodically with a frequency of $\nu$
• B. changes periodically with a frequency of $2\nu$
• C. changes periodically with a frequency of $\frac{\nu }{2}$
• D. remains constant

Answer 1

The correct option is B changes periodically with a frequency of $2\nu$

The frequency with which the particle executing $SHM$ oscillates is $\nu$

$\therefore$ $w=2\pi \nu$

The displacement equation of the particle

$x=Asinwt$ OR $x=Asin\left(2\pi \nu t\right)$

Velocity of the particle $v=w\sqrt{A^2-x^2}$

$\therefore$ $v^2=w^2(A^2-x^2)=w^2[A^2-A^{2}si{n}^2(2\pi \nu t\left)\right]$

$⟹$ $v^2=A^2{w}^{2}co{s}^2\left(2\pi \nu t\right)$ $(\because 1-si{n}^2\theta =co{s}^2\theta )$ or $v^2=\frac{A^2{w}^2}{2}[1+cos(4\pi \nu t\left)\right]$ $(\because 1+cos2\theta =2co{s}^2\theta )$

Kinetic energy of the particle: $E=\frac{1}{2}m{v}^2=\frac{1}{4}m{w}^2{A}^2[1+cos(4\pi \nu t\left)\right]$

Thus, the Kinetic energy of the particle oscillates with frequency $2\nu$.