Q. If a simple pendulum of length, $L$ has maximum angular displacement $\alpha$, then the maximum kinetic energy of bob of mass $M$ is
A) $\frac{1}{2}\ \frac{ML}{g}$
B) $\frac{Mg}{2L}$
C) $MgL(1 - \cos{\alpha)}$
D) $\frac{MgL\ \sin\ \alpha}{2}$
Explanation
(c) Given, mass of bob=$M$
Length of simple pendulum=$L$
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Now, in $\mathrm{\Delta}OAB,\ \ \cos{\alpha = \frac{OA}{OB}}$
or $\cos{\alpha = \frac{OA}{L}}$
or $AO = L\ \cos\ \alpha$
and $AP = OP - AO$
$AP = L - L\ \cos\ \alpha$
or $h = L(1 - \cos\ \alpha)$
The maximum kinetic energy of bob
= Maximum potential energy
$KE = Mgh$
$KE = MgL(1 - \cos\ \alpha$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.