A body is performing SHM. Then its

see full answer

Start JEE / NEET / Foundation preparation at rupees 99/day !!

21% of IItians & 23% of AIIMS delhi doctors are from Sri Chaitanya institute !!

An Intiative by Sri Chaitanya

all are true

root mean square velocity is $\large \frac{1}{{\sqrt 2 }}$ times of its maximum velocity

average total energy per cycle is equal to its maximum kinetic energy

average kinetic energy per cycle is equal to half of its maximum kinetic energy

answer is D.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Detailed Solution

Suppose equation of SHM is $\large x\, = \,A\cos \omega t$

$\large \therefore \,v\, = \,\frac{{dx}}{{dt}}\, = \, - A\omega \sin \omega t$

$\large \therefore \,{v_{\max }}\, = \,A\omega$

$\large \int\limits_0^T {{v^2}dt} \, = \,\int\limits_0^{\frac{{2\pi }}{\omega }} {{A^2}{\omega ^2}{{\sin }^2}\omega tdt\, = \,\pi }$

$\large \therefore \;{V_{rms}}\, = \,\sqrt {\frac{{\int\limits_0^T {{v^2}dt} }}{T}} \,\, = \,\frac{{A\omega }}{{\sqrt 2 }}\, = \,\frac{{{V_{\max }}}}{{\sqrt 2 }}$

$\large {\left( {KE} \right)_{\max }}\, = \,\frac{1}{2}mv_{\max }^2\, = \,\frac{1}{2}m{A^2}{\omega ^2}$

$\large PE\, = \,\frac{1}{2}m{\omega ^2}{x^2} = \,\frac{1}{2}m{\omega ^2}{A^2}co{s^2}\omega t$

$\large KE\, = \,\frac{1}{2}m{v^2}$

$\large KE\, = \,\frac{1}{2}m{A^2}{\omega ^2}{\sin ^2}\omega t$

$\large \therefore$ Total energy = KE+PE = $\large \frac{1}{2}m{v^2}{A^2}$

so option - (4) is right.

Watch 3-min video & get full concept clarity

tricks from toppers of Infinity Learn

Result Producing online coaching for JEE/NEET of south India

Largest All India Test Series for JEE/NEET

Best Books for IIT JEE

Best Books for NEET

🔥 Start Your JEE/NEET Prep at Just ₹1999 / month - Limited Offer! Check Now!

Get Expert Academic Guidance – Connect with a Counselor Today!

Result Producing online coaching for JEE/NEET of south India

Largest All India Test Series for JEE/NEET

Best Books for IIT JEE

Best Books for NEET