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)

I and II are correct

Answer 2

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