Question

Consider the following statements :
The total energy of a particle executing simple harmonic motion depends on its :
(I) amplitude
(II) period
(III) displacement Of these statements :

• A. I and II are correct
• B. II and III are correct
• C. I and III are correct
• D. I, II and III are correct

Answer 1

Answer: (A)

Key Idea: Total energy of a particle executing simple harmonic motion is obtained by summing its potential and kinetic energies.
Potential energy of particle in SHM
$U=\frac{1}{2}m{\omega }^2{x}^2$
or $U=\frac{1}{2}m{(2\pi f)}^2{x}^2$
or $U=2{\pi }^{2}m{f}^2{x}^2$ ?(i)
Kinetic energy of particle in SHM
$K=\frac{1}{2}m{\omega }^2(A^2-x^2)$
or $K=2{\pi }^{2}m{f}^2(A^2-x^2)$ ?(ii)
Hence, total energy
$E=K+U$
$=2{\pi }^{2}m{f}^2{x}^2+2{\pi }^{2}m{f}^2(A^2-x^2)$
$=2{\pi }^{2}m{f}^2{A}^2=\frac{2{\pi }^{2}m{A}^2}{T^2}$
$\left(\because T=\frac{1}{f}\right)$
Thus, it is obvious that total energy of particle executing simple harmonic motion depends on amplitude (A) and period (T). $II$ and $III$ are correct