Simple Pendulum Kinetic Energy vs. Time Graph Analysis

Understanding Kinetic Energy in Simple Harmonic Motion

For a simple pendulum undergoing Simple Harmonic Motion (SHM), the kinetic energy ($K.E.$) varies cyclically with time ($t$). Key characteristics of this variation are:

• $K.E.$ is maximum at the mean (equilibrium) position where the pendulum's speed is greatest.
• $K.E.$ is zero at the extreme positions (maximum displacement) where the pendulum's speed momentarily drops to zero.
• Since $K.E. = \frac{1}{2} m v^2$, it is always non-negative ($K.E. \ge 0$).
• The kinetic energy pattern repeats every half of the time period ($T/2$). This means the frequency of $K.E.$ variation is double the frequency of the pendulum's oscillation. The mathematical form is often related to $\sin^2(\omega t)$ or $\cos^2(\omega t)$.

Analyzing the K.E. vs. Time Graph

The correct graph representing $K.E.$ versus $t$ must satisfy these conditions:

• Non-Negative Values: The graph must always lie above or on the time axis ($K.E. \ge 0$). Graphs showing negative $K.E.$ are incorrect.
• Zero Points: $K.E.$ should be zero at times corresponding to the extreme positions of the pendulum.
• Maximum Points: $K.E.$ should reach its peak value at times corresponding to the mean position.
• Periodicity: The energy cycle (e.g., from zero to maximum and back to zero) should repeat twice within one full oscillation period ($T$), indicating a period of $T/2$ for the $K.E.$ variation itself.

Considering these points, if the pendulum starts at an extreme position ($t=0$), its $K.E.$ is $0$. It increases to a maximum at $t=T/4$ (mean position), decreases to $0$ at $t=T/2$ (other extreme), increases to a maximum again at $t=3T/4$, and returns to $0$ at $t=T$. Option B accurately displays this behavior, showing a wave that starts at zero, rises to a peak, falls back to zero, and repeats this pattern, staying entirely within the non-negative domain and exhibiting the required periodicity.