Question

The figure below shows a graph of potential energy $U\left(x\right)$ verses position $x$ for a particle executing one dimentional motion along the x-axis. The total mechanical energy of the system is indicated by the dashed line. Choose the correct statement for the position of particle varying between points $A$ and $G$.

• A. The magnitude of force is maximum at $D$
• B. The kinetic energy is maximum at $B$
• C. The velocity is zero at $A$ and $G$
• D. None of the above

Answer 1

The correct option is C The velocity is zero at $A$ and $G$

From relation between potential energy and force:
$F\left(x\right)=\frac{-dU}{dx}=-$(Slope of $U-x$ curve)

$\Rightarrow$Magnitude of force is maximum where the magnitude of slope is maximum hence $A$ option is false, since at $D$ slope = 0

$\Rightarrow$$K{E}_{\text{max}}$ will occur at a point if $PE=0$ and net force is zero i.e $F=0$

It will hold for $F=0$, because the acceleration $a=0$
$\therefore \frac{dv}{dt}=0$
Therefore velocity is maximised and $K{E}_{\text{max}}$ will exist at that point.
$\therefore \text{at}D$, $K{E}_{\text{max}}$ will occur.

$\Rightarrow$Considering above points motion of particle will be represented as:

So we can conclude that particle must be experiencing a conservative restoring force ($F$), hence it's velocity will be zero at points $A$ and $G$, where it's potential energy is
maximum.
$\therefore$ option $\left(c\right)$ is correct