Question

A body of mass $m$ is attached to a spring of spring constant $k$ which hangs from the ceiling of an elevator at rest in equilibrium. Now the elevator starts accelerating upwards with its acceleration varying as $a=pt+q$, where $p$ and $q$ are positive constants. In the frame of the elevator

• A. The block will perform SHM for all values of $p$ and $q$.
• B. The block will not perform SHM in general for all values of $p$ and $q$ except $p$ = 0.
• C. The block will perform SHM for all values of $p$ and $q$ except $p=0$.
• D. The velocity of the block will vary simple harmonically for all values of $p$ and $q$.

Answer 1

Answer: (B), (D)

The correct options are
B The block will not perform SHM in general for all values of $p$ and $q$ except $p$ = 0.
D The velocity of the block will vary simple harmonically for all values of $p$ and $q$.

In the frame of elevator
$mg+ma-kx=m\frac{d^{2}x}{d{t}^2}\Rightarrow \frac{d^{2}x}{d{t}^2}=-\frac{k}{m}[x-\frac{m(g+a)}{k}]\Rightarrow \frac{d^{2}x}{d{t}^2}=-\frac{k}{m}[x-\frac{m(g+pt+q)}{k}]$
There is a term involving $t$ on R.H.S ., this does not represent S.H.M. unless $p=0$.
Differentiating again w.r.t. time.
$\frac{d^{3}x}{d{t}^3}=-\frac{k}{m}[\frac{dx}{dt}-\frac{mp}{k}]or\frac{d^{2}v}{d{t}^2}=-\frac{k}{m}[v-\frac{mp}{k}]$
Thus the velocity of the block will vary simple harmonically.