First slide

Simple harmonic motion

Question

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

Difficult

A

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

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

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

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

Solution

$\begin{array}{l} a x^{2} + b v^{2} = k \\ b v^{2} = k - a x^{2} \\ v^{2} = \frac{k}{b} - \frac{a}{b} x^{2} \end{array}$
$Compare with v^{2} = A^{2} ω^{2} - ω^{2} x^{2}$
$\begin{array}{l} ω^{2} = a / b and A^{2} ω^{2} = k / b \\ A = \sqrt{\frac{A^{2} ω^{2}}{ω^{2}}} = \sqrt{\frac{k / b}{a / b}} = \sqrt{\frac{k}{a}} \end{array}$

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