First slide

Simple harmonic motion

Question

The total energy of the body executing S.H.M. is E. Then the kinetic energy when the displacement is half of the amplitude, is

Moderate

A

$\frac{E}{2}$
B

$\frac{E}{4}$
C

$\frac{3 E}{4}$
D

$\frac{\sqrt{3}}{4} E$

Solution

Total energy in SHM $E = \frac{1}{2} {mω}^{2} a^{2}$ ; (where a = amplitude)
Potential energy
$KE = \frac{1}{2} {mω}^{2} (a^{2} - y^{2}) U = \frac{1}{2} {mω}^{2} y^{2}$
When $y = \frac{a}{2}$
$\Rightarrow KE = E - U K E = E - \frac{1}{2} {mω}^{2} (\frac{a^{2}}{4}) = E - \frac{E}{4} = \frac{3 E}{4}$

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