Starting from the mean position,a body oscill | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) The kinetic energy $(KE)$ of a particle in simple harmonic motion is given by $KE = \frac{1}{2} k A^2 \cos^2(\omega t)$,where $A$ is the amplitude

Vedclass · Accepted Answer

$\frac{T}{12}$

Vedclass · Answer

(B) The kinetic energy $(KE)$ of a particle in simple harmonic motion is given by $KE = \frac{1}{2} k A^2 \cos^2(\omega t)$,where $A$ is the amplitude and $\omega$ is the angular frequency.Given that the body starts from the mean position,the displacement is $x = A \sin(\omega t)$,and the velocity is $v = A \omega \cos(\omega t)$.The total energy $(TE)$ is $\frac{1}{2} k A^2$.We are given $KE = 75 \% 	ext{ of } TE = \frac{3}{4} TE$.So,$\frac{1}{2} k (A \cos(\omega t))^2 = \frac{3}{4} (\frac{1}{2} k A^2)$.$\cos^2(\omega t) = \frac{3}{4}$.$\cos(\omega t) = \frac{\sqrt{3}}{2}$.Since $\cos(30^{\circ}) = \frac{\sqrt{3}}{2}$,we have $\omega t = 30^{\circ} = \frac{\pi}{6} 	ext{ radians}$.Substituting $\omega = \frac{2\pi}{T}$,we get $\frac{2\pi}{T} t = \frac{\pi}{6}$.Solving for $t$,we get $t = \frac{T}{12}$.

Starting from the mean position,a body oscillates simple harmonically with a period $T$. After what time will its kinetic energy be $75 \%$ of the total energy? $(\sin 30^{\circ} = 0.5)$

Similar Questions

On what factors does the mechanical energy of a simple harmonic oscillator depend,and on what factors does it not depend?

In a simple harmonic oscillator,at the mean position:

For a particle executing simple harmonic motion,the ratio of kinetic and potential energies at a point where displacement is one half of the amplitude is

The $P.E.$ of a particle executing $SHM$ at a distance $x$ from its equilibrium position is

$A$ particle starting from mean position executes simple harmonic motion with a period $8 \ s$. The minimum time in which its potential energy becomes half of the total energy is . . . . . . . (in $s$)

Vedclass Test Series

Exam Paper Generator

Online Exam Module