First slide

Simple harmonic 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 first time. The equation of motion of the particle can be represented by

Moderate

A

$x = 3 + 3 \cos π t$
B

$x = 3 \cos π t$
C

$x = 3 + 3 \sin π t$
D

$x = 3 + 3 \cos (2 π t)$

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 r - 0 is 0.5 second, that is, T/2=0.5 sec, hence T=1 second. Hence the equation of motion is $x = 3 + A (\sin ω t + ϕ_{0}$ ) where $A = 3, ω = 2 π \times 1$ and $ϕ_{0} = \frac{π}{2} . So x = 3 + 3 \cos (2 π t)$

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