Question

1. Find the total energy of particle executing SHM? 

2. Show graphically the variation of PE and KE with time In SHM. 

3. What is the frequency of these energies w.r.t the frequency of the particle executing SHM?

Answer 1

1. PE at any instant

PE = \(\frac{1}{2}\) kx²

= \(\frac{1}{2}\)mω² A² sin² ωt

(∵ k = mω² and x = A sin ωt)

⇒ PE = \(\frac{1}{2}\) mω² A² sin² ωt

KE at any instant KE = \(\frac{1}{2}\) mv²

KE = \(\frac{1}{2}\) mA² ω² cos² ωt

Total energy = KE + PE

⇒ Total energy = \(\frac{1}{2}\) mω² A²sin² ωt + \(\frac{1}{2}\) mω² A² cos² ωt 

⇒ Total energy = \(\frac{1}{2}\) mω² A²

2. Graph

3. Frequency = \(\frac{2}{period} = \frac{2}{T}\).