First slide

Simple hormonic motion

Question

A particle is executing SHM according to the equation x = A cos $ωt$ . Average speed of the particle during the interval $0 \leq t \leq \frac{π}{6 ω}$

Moderate

A

$\frac{\sqrt{3} Aω}{2}$
B

$\frac{\sqrt{3} Aω}{4}$
C

$\frac{3 Aω}{π}$
D

$\frac{3 Aω}{π} (2 - \sqrt{3})$

Solution

Average velocity $v = \frac{\int_{0}^{t} \frac{dx}{dt} . dt}{t} = \frac{\int_{0}^{t} dx}{t} = \frac{x (t) - x (0)}{t}$
= $\frac{A (\cos \frac{π}{6} - 1)}{\frac{π}{6 ω}} = \frac{3 Aω}{π} (\sqrt{3} - 2)$
since particle does not change its direction in the given interval , average speed
$| v | = \frac{3 Aω}{π} (2 - \sqrt{3})$

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