For a simple pendulum, a graph is plotted between its kinetic energy (K.E.) and potential energy (P.E.) against its displacement d. which one of the following represents these correctly? (graphs are schematic and not drawn to scale)

• A

  
• B

  
• Correct answer
  C

  
• D

  

Solution

For simple pendulum performing the simple harmonic motion, displacement, $\mathit{\text{y}}=A\sin \mathit{\text{ωt}}$$v=\frac{dy}{dt}=\mathit{\text{ωA}}cos\mathit{\text{ωt}}\mathit{=}\mathit{\text{ω}}\sqrt{{\mathit{\text{A}}}^{\mathit{2}}\mathit{-}{\mathit{\text{y}}}^{\mathit{2}}}$$\mathit{\text{⇒K.E}}\text{.}=\frac{1}{2}{\mathit{mv}}^2$, 
This represents a downward parabola, $Y=\pm A$, $\mathit{\text{K.E.}}=\frac{1}{2}{\mathit{\text{ω}}}^2\mathit{\text{m}}\left({\mathit{\text{A}}}^2-{\mathit{\text{A}}}^2\right)=0$. $\mathit{\text{P.E.}}=\frac{1}{2}{\mathit{\text{mω}}}^2{\mathit{\text{y}}}^2$. This represents an upward parabola. On plotting graphs of potential energy & kinetic energy.

Hence, the correct answer is option: 'C'.