Question

The total energy of a particle, executing simple harmonic motion is:

• A. $\propto x$
• B. $\propto x^2$
• C. independent of $x$
• D. $\propto x^{1/2}$

Answer 1

The correct option is C independent of $x$

For a particle in simple harmonic motion, $x\left(t\right)=Acos(\omega t+\varphi )$. Therefore,

Potential energy, $U=\frac{1}{2}k{x}^2=\frac{1}{2}k{A}^{2}co{s}^2(\omega t+\varphi )$

Kinetic energy, $K.E=\frac{1}{2}m{v}^2=\frac{1}{2}m(\frac{dx}{dt}{)}^2=\frac{1}{2}k{A}^{2}si{n}^2(\omega t+\varphi )$

Now, Total energy

$E=U+K.E$

$=\frac{1}{2}k{A}^{2}co{s}^2(\omega t+\varphi )+\frac{1}{2}k{A}^{2}si{n}^2(\omega t+\varphi )=\frac{1}{2}k{A}^2$

Therefore, total energy is independent of x.