Question

A particle starts Simple harmonic motion from the mean position. Its amplitude is a and total energy E. At on instant its kinetic energy is $\frac{3E}{4}.$ Its displacement at that instant is

• A. $\frac{a}{\sqrt{2}}$
• B. $\frac{a}{2}$
• C. $\sqrt{3}\frac{a}{2}$
• D. zero

Answer 1

The correct option is B $\frac{a}{2}$

The equation of the simple harmonic motion is $y=Asinwt$ with usual notations$.$

The total energy is $\left(\frac{1}{2}\right)m{w}^2{a}^2$ and the kinetic energy is $\left(\frac{1}{2}\right)m{w}^2\left(a^2-y^2\right)$

therefore we have $\left(\frac{1}{2}\right)m{w}^2{a}^2=E$ and

$\left(\frac{1}{2}\right)m{w}^2\left(a^2-y^2\right)=\frac{3E}{4}$

Dividing the second equation by the first$,$ $1-\frac{y^2}{a^2}=\frac{3}{4},$ from which $y=\frac{a}{2}.$

Hence,

option $\left(B\right)$ is correct answer.