If the kinetic energy of simple pendulum is 4 1 of its total energy, then the displacement ( x ) and amplitude ( A ) are related as

Question

If the kinetic energy of simple pendulum is 4 1 ​ of its total energy, then the displacement ( x ) and amplitude ( A ) are related as

TestHub · Accepted Answer

Total energy of a simple pendulum executing SHM is given by $E = \frac{1}{2} m ω^{2} A^{2}$

Kinetic energy at displacement $x$ is given by $K = \frac{1}{2} m ω^{2} (A^{2} - x^{2})$

Given that $K = \frac{1}{4} E$

$\frac{1}{2} m ω^{2} (A^{2} - x^{2}) = \frac{1}{4} (\frac{1}{2} m ω^{2} A^{2})$

$A^{2} - x^{2} = \frac{A ^{2}}{4}$

$x^{2} = A^{2} - \frac{A ^{2}}{4} = \frac{3 A ^{2}}{4}$

$x = \frac{3}{2} A$

Answer: $x = \frac{3}{2} A$

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