Question

A particle executes S.H.M (a) What fraction of total energy is kinetic? What fraction is potential when displacement is one half of the amplitude?

• A. $\frac{3}{4}$ and $\frac{1}{4}$
• B. $\frac{2}{3}$ and $\frac{1}{3}$
• C. $\frac{1}{2}$ and $\frac{1}{2}$
• D. $\frac{1}{6}$ and $\frac{5}{6}$

Answer 1

The correct option is A

$\frac{3}{4}$ and $\frac{1}{4}$

The total energy of a particle doing SHM is $\frac{1}{2}K{A}^2$. NOw at any displacement x from the mean position the P,E is $\frac{1}{2}k{x}^2$ so at this position K.E = T.E - P.E $=\frac{1}{2}K{A}^2-\frac{1}{2}k{x}^2$
When displacement is half the amplitude
x = $\frac{A}{2}$
$\therefore P.E=\frac{1}{2}k{\left(\frac{A}{2}\right)}^2\frac{1}{4}K{A}^2=\frac{1}{4}xT.EK.E.=\frac{1}{2}k{A}^2-\frac{1}{4}x\frac{1}{2}K{A}^2=\frac{3}{4}x\frac{1}{2}K{A}^2=\frac{3}{4}xT.E$
So the correct option is (a).