Question

In case of SHM what fraction of total energy is kinetic and what fracton is potential, when displacement is one fourth of amplitude.

Answer 1

The total energy is $E=\frac{1}{2}k{x}^2$

The potential energy

$U=\frac{1}{2}k{x}^2$

When displacement is one fourth of amplitude

$U=\frac{1}{2}k{\left(\frac{x}{4}\right)}^2$

$U=\frac{1}{2}k\frac{x^2}{16}$

$U=\frac{E}{16}$

Now, the percentage of potential energy is

$U=\frac{E}{16}\times 100$

$U=6.25$ %

The potential energy is $6.25$ % of $E$

Now, the kinetic energy is

$K=E-U$

$K=E-\frac{E}{16}$

$K=\frac{15E}{16}$

Now, the percentage of kinetic energy

$K=\frac{15E}{16}\times 100$

$K=15\times 6.25$

$K=93.75$ %

The kinetic energy is $93.75$ % of $E$

Hence, the percentage of potential energy to the total energy will be $6.25$ and the percentage of kinetic energy to the total energy will be $93.75$ in the simple harmonic motion