First slide

Simple hormonic motion

Question

A particle is moving along the x-axis under the influence of a force given by $F = - 5 x + 15$ . 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 mass of the particle is

Moderate

A

$3 π^{2}$
B

$\frac{5 π^{2}}{4}$
C

$\frac{5}{4 π^{2}}$
D

$\frac{1}{3 π^{2}}$

Solution

A particle moving in a SHM about $x = 3$ with amplitude 3 will exhibit a motion similar to that described in the problem.
Hence $x = 3 + 3 \cos (ω t)$ .
Also $\frac{T}{2} = 0.5 \Rightarrow T = 1 \Rightarrow ω = 2 π$ .
Therefore force $F = m \frac{d^{2} x}{d t^{2}} = - m ω^{2} (x - 3) = - m ω^{2} x + 3 m ω^{2}$ .
Comparing with the equation given in the problem $m ω^{2} = 5 \Rightarrow m = \frac{5}{4 π^{2}}$ .

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