Question

A particle is moving along the $x$ -axis under the innuence 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 equation of motion of the particle can be represented by

• A. $x=3+3 \cos \pi t$
• B. $x=3 \cos \pi t$
• C. $x=3+3 \sin \pi t$
• D. $x=3+3 \cos (2 \pi t)$

Answer 1

Answer: (D)

The mean position of particle can be found by setting
$F=-5 x+15=0$,
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 $T / 2=0.5\, sec$,
hence $T=1$ second.
Hence the equation of motion is
$x=3+A\left(\sin \omega t+\phi_{0}\right)$
where $A=3, \omega=2 \pi \times 1$ and
$\phi_{0}=\frac{\pi}{2} .$
So $x=3+3 \cos (2 \pi t)$