Question

The potential energy of a particle of mass $0.1kg$ moving along the $x-$axis is given by $U=5x(x–4)J$, where $x$ is in metre. It can be concluded that

• A. the particle is acted upon by the constant force
• B. the speed of particle is maximum at $x=2m$
• C. the particle cannot execute $SHM$
• D. the period of oscillation of particle is $\frac{\pi }{20}s$

Answer 1

The correct option is B the speed of particle is maximum at $x=2m$

Relation between Potential energy and coservative forces $F=-\frac{dU}{dx}=-[5\times 2x-20]=-10x+20a=\frac{F}{m}=-100x+200a=-100(x-2)=-100X$

Where, $X=x-2$ and ${\omega }^2=100\Rightarrow \omega =10$

As $a\propto X$, the particle performs $SHM$

At mean position, $X=0$, i.e. $x=2m$, so speed of the particle is maximum at $x=2m$.

Time period of oscillation is given by,

$\omega =\frac{2\pi }{T}=10\Rightarrow T=\frac{\pi }{5}s$