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