First slide

Simple hormonic motion

Question

A particle moves along a straight line to follow the equation ${ax}^{2} + {bv}^{2} = k$ , where a, b and k are constants and x and v are x-coordinate and velocity of the particle respectively. Find the amplitude.

Moderate

A

$\sqrt{\frac{k}{b}}$
B

$\sqrt{\frac{b}{k}}$
C

$\sqrt{\frac{a}{k}}$
D

$\sqrt{\frac{k}{a}}$

Solution

${ax}^{2} + {bv}^{2} = k$
${bv}^{2} = k - {ax}^{2}$
$v^{2} = \frac{k}{b} - \frac{a}{b} x^{2}$
Compare with $v^{2} = A^{2} ω^{2} - ω^{2} x^{2}$
$ω^{2} = \frac{a}{b} and A^{2} ω^{2} = \frac{k}{b}$
$A = \sqrt{\frac{A^{2} ω^{2}}{ω^{2}}} = \sqrt{\frac{\frac{k}{b}}{\frac{a}{b}}} = \sqrt{\frac{k}{a}}$

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