For a simple pendulum, having time period T, the variation of kinetic energy (K.E.) with time (t) is represented by: ____.
Show Hint
Energy graphs in SHM never go below the time axis. Total energy is a flat horizontal line, while K.E. and P.E. are bell-shaped curves that swap values as the pendulum swings.
Step 1: Understanding the Concept:
Kinetic Energy (K.E.) is proportional to the square of the velocity ($v^2$). In Simple Harmonic Motion (SHM), velocity is a sine or cosine function of time. Step 2: Key Formula or Approach:
1. Velocity $v = \omega A \cos(\omega t)$ (if starting from equilibrium)
2. $K.E. = \frac{1}{2}mv^2 = \frac{1}{2}m\omega^2A^2 \cos^2(\omega t)$ Step 3: Detailed Explanation:
1. Positivity: Since K.E. depends on $v^2$, it is always positive or zero; it never goes negative.
2. Frequency: The K.E. fluctuates twice during one full period $T$ of the pendulum (once at each pass through the equilibrium point). Therefore, its period is $T/2$.
3. Shape: It follows a $\sin^2$ or $\cos^2$ shape, appearing as a series of positive "humps." Step 4: Final Answer:
The correct graph is a periodic, non-negative wave with twice the frequency of the displacement.
