Question

A particle starts executing simple harmonic motion $\left(\text{SHM}\right)$ of amplitude ${}^{\prime }{a}^{\prime }$ and total energy $E$. At any instant, its kinetic energy is $\frac{3E}{4}$ then its displacement ${}^{\prime }{y}^{\prime }$ is given by :

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

Answer 1

The correct option is A $y=\frac{a}{2}$

Energy for a particle performing $\text{SHM}$ is given by,

$E=\frac{1}{2}K{a}^2.......\left(i\right)$

Since, for a given instance $KE=\frac{3E}{4}$

$\frac{3E}{4}=\frac{1}{2}K(a^2-y^2)$

$\frac{3}{4}\times \frac{1}{2}K{a}^2=\frac{1}{2}K(a^2-y^2)$

$y^2=a^2-\frac{3a^2}{4}=\frac{a^2}{4}$

$\therefore y=\pm \frac{a}{2}$

Hence, option $\left(\text{D}\right)$ is correct.