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 Topic:
This question relates to "Oscillations" and the energy transformations within Simple Harmonic Motion (SHM). A simple pendulum continuously converts potential energy into kinetic energy and vice versa. The graph of these energies over time provides insight into the frequency of energy exchange compared to the frequency of the displacement. Step 2: Key Formulas and Approach:
Velocity in SHM: $v(t) = v_0 \cos(\omega t + \phi)$.
Kinetic Energy: $K.E. = \frac{1}{2} m v^2$.
Relationship: $K.E.(t) = \frac{1}{2} m v_0^2 \cos^2(\omega t + \phi)$.
Step 3: Detailed Explanation:
Non-negativity: Kinetic energy is proportional to the square of velocity. Because any real number squared is non-negative, the graph of K.E. must always be on or above the time axis. It never goes into the negative region.
Frequency Doubling: In one full oscillation (period $T$) of the pendulum, the bob passes through the equilibrium point twice (once in each direction). Since K.E. is maximum at equilibrium, it reaches its peak twice per cycle. This means the period of the K.E. variation is $T/2$, and its frequency is double the pendulum's frequency.
Waveform Shape: The function $\cos^2(\theta)$ or $\sin^2(\theta)$ creates a series of smooth "humps."
Analysis of options: We look for a graph that is purely positive, periodic, and shows two peaks within the duration of one standard time period $T$. Usually, Plot 3 or Plot 2 in standard textbooks shows this rectified-sine-wave appearance.
Step 4: Final Answer:
The correct graph is a periodic, non-negative wave with a frequency twice that of the displacement.
