First slide

Simple hormonic motion

Question

A particle is moving along the x-axis under the influence of a force given by F = -5x + 15. At time t = 0, the particle is located at x = 6 and is having zero velocity. It takes 0.5 seconds to reach the origin for the fist time. The equation of motion of the particle can be represented by

Moderate

A

x = 3 + 3 cos $πt$
B

x = 3 cos $πt$
C

$x = 3 + 3 \sin πt$
D

$x = 3 + 3 \cos (2 πt)$

Solution

The mean position of particle can be found by setting F = -5x + 15 = 0, so the mean position lies at x = 3. One extreme position is at x = 6. Hence the other extreme position for this particle undergoing SHM should be at x = 0. Time taken by particle to reach from x = 6 to x = 0 is 0.5 second, that is $\frac{T}{2}$ = 0.5 sec, hence T = 1 second. Hence the equation of motion is $x = 3 + A \sin (ω t + \emptyset_{0})$ , where A = 3, $ω = \frac{2 π}{1} = 2 π$ and $\emptyset_{0} = \frac{π}{2} . So x = 3 + 3 \cos (2 πt)$ .

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