In the following table, the displacement of a $Simple \text{ } Harmonic \text{ } Oscillator \text{ } (SHO)$ is shown in column-$I$ and the kinetic energy is shown in column-$II$. Match them appropriately.
Column-$I$Column-$II$
$(a)$ $y = \frac{A}{\sqrt{2}}$$(i)$ $K = \frac{3E}{4}$
$(b)$ $y = \frac{\sqrt{3}A}{2}$$(ii)$ $K = \frac{E}{4}$
$(iii)$ $K = \frac{E}{2}$

Vedclass pdf generator app on play store
Vedclass iOS app on app store
(A) The total energy of a $Simple \text{ } Harmonic \text{ } Oscillator$ is $E = \frac{1}{2}kA^2$.
The potential energy at displacement $y$ is $U = \frac{1}{2}ky^2$.
The kinetic energy is $K = E - U = \frac{1}{2}k(A^2 - y^2)$.
For $(a)$ $y = \frac{A}{\sqrt{2}}$:
$K = \frac{1}{2}k(A^2 - \frac{A^2}{2}) = \frac{1}{2}k(\frac{A^2}{2}) = \frac{1}{2}(\frac{1}{2}kA^2) = \frac{E}{2}$. Thus, $(a) - (iii)$.
For $(b)$ $y = \frac{\sqrt{3}A}{2}$:
$K = \frac{1}{2}k(A^2 - \frac{3A^2}{4}) = \frac{1}{2}k(\frac{A^2}{4}) = \frac{1}{4}(\frac{1}{2}kA^2) = \frac{E}{4}$. Thus, $(b) - (ii)$.
Therefore, the correct matching is $(a-iii, b-ii)$.

Explore More

Similar Questions

$A$ mass of $1 \text{ kg}$ is executing $SHM$. Its displacement is given by $x = 6.0 \cos(100t + \pi/4) \text{ cm}$. What is the maximum kinetic energy (in $text{ J}$)?

Starting from the origin,a particle oscillates simple harmonically with a time period of $2 \ s$. After what time will its kinetic energy be $75 \%$ of the total energy?

$A$ body performing simple harmonic motion has potential energy $P_{1}$ at displacement $x_{1}$. Its potential energy is $P_{2}$ at displacement $x_{2}$. The potential energy $P$ at displacement $(x_{1}+x_{2})$ is

An object of mass $0.2 \ kg$ executes simple harmonic oscillations along the $X$-axis with a frequency of $(\frac{25}{\pi}) \ Hz$. At the position $x=0.04 \ m$,the object has kinetic energy $1 \ J$ and potential energy $0.6 \ J$. The amplitude of oscillation is (in $m$)

$A$ particle is executing simple harmonic motion $(SHM)$ of amplitude $A$ along the $x$-axis about $x = 0$. When its potential energy $(PE)$ equals kinetic energy $(KE)$,the position of the particle will be

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial
For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free
For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo