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Step-by-Step Explanation
Step 1: Recognize Simple Harmonic Motion (SHM)
A simple pendulum executing small oscillations behaves like a simple harmonic oscillator (SHO). If its maximum angular displacement is small, the restoring force is proportional to the displacement, leading to SHM.
Step 2: Express Potential Energy (PE)
In SHM, the potential energy (PE) of the bob (assuming displacement measured from the mean position as d) can be written as:
$PE = \frac{1}{2}k\,d^2$
where $k$ is a constant related to the effective stiffness in SHM. Notice that at the mean position (d = 0), the potential energy is zero, while at the extreme positions (d = A), the potential energy is maximum.
Step 3: Express Kinetic Energy (KE)
The total energy $E$ of the pendulum in SHM is constant and given by:
$E = \frac{1}{2}k\,A^2
At a displacement d from the mean position, the kinetic energy (KE) is the remaining part of total energy after subtracting the potential energy. Hence,
$KE = E - PE = \frac{1}{2}k\,A^2 - \frac{1}{2}k\,d^2 \;=\; \frac{1}{2}k\,(A^2 - d^2).
This means KE is maximum at the mean position (where $d = 0$) and zero at the extremes (where $d = \pm A$).
Step 4: Identify the Correct Graph
PE vs. d is a parabola opening upwards, starting at zero when $d = 0$ and increasing with $d^2$.
KE vs. d is a downward-opening parabola, maximum at $d = 0$ and zero at $d = \pm A$.
The correct schematic representation (as indicated by the provided correct answer image) shows these parabolic variations of KE and PE with displacement.
Conclusion
Thus, the correct graph is the one in which potential energy curves upward with increasing $|d|$, while kinetic energy curves downward, being largest at $d = 0$ and zero at the extremes.
