Question

The displacement $\left(x\right)$ of a particle as a function of time $\left(t\right)$ is given by
where $a,b$ and $c$ are constant of motion. Choose the correct statements from the following.

• A. The motion repeats itself in a time interval of $2\pi /b$
• B. The energy of the particle remains constant.
• C. The velocity of the particle is zero $x=\pm a$
• D. The acceleration of the particle is zero at $x=\pm a$

Answer 1

Answer: (A), (B), (C)

The correct options are
A The motion repeats itself in a time interval of $2\pi /b$
B The energy of the particle remains constant.
C The velocity of the particle is zero $x=\pm a$

The motion of the particle is simple harmonic.
The displacement at time $t$ is $x=a\sin (bt+c)$
Therefore, displacement at time $(t+(2\pi /b\left)\right)$ is
$x$ at $(t+\frac{2\pi }{b})=a\sin [b(t+\frac{2\pi }{b})+c]$
$=a\sin [bt+c+2\pi ]=a\sin (bt+c)=x$ ( at time $t$)
Hence, statement (a) is correct.
Statement (b) is also correct since the same displacement is recovered after a time interval of $\left(2\pi /b\right)$. Statement (c) is correct because the velocity is zero when the displacement = $\pm$ the amplitude, i.e at the extreme ends of the motion. Statement (d) is incorrect, the acceleration is maximum ( in magnitude ) at $x=\pm A$.