Question

A body is in motion along the positive $x$ - axis, according to the relation $x=a{\sin }^2\omega t$. Then, the variation of its kinetic energy $K$ with time $t$ may be represented by

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

The instantaneous velocity of the body is
$v=\frac{dx}{dt}$
$=\frac{d}{dt}\left({\text{a sin}}^2\omega t\right)$
$=2a\omega \text{sin}\omega t\text{cos}\omega t$
$=a\omega \text{sin}2\omega t$
Let the mass of the body is $m$.
The expression for the kinetic energy of the body is
$K=\frac{1}{2}m{v}^2$
Substitute the known values.
$K=\frac{1}{2}m(a\omega \text{sin}2\omega t{)}^2$
$=\frac{1}{2}m{a}^2{\omega }^2{\text{sin}}^{2}2\omega t......\left(1\right)$
$K=\frac{1}{2}m{a}^2{\omega }^2\frac{(1-\cos 4\omega t)}{2}$
The time period of the expression of kinetic energy is $\frac{2\pi }{4\omega }=\frac{\pi }{2\omega }$
The maxima of the equation of kinetic energy occurs at $\frac{\pi }{4\omega }$.
The kinetic energy will be zero at $t=0$.
Hence option d is satisfying the equation of kinetic energy.