$A$ particle is released from rest at the origin. It moves under the influence of a potential field $U = x^2 - 3x$. The kinetic energy at $x = 2$ is ............... $J$.

The work done by the force $\vec{F} = x^2 \hat{i} + y^2 \hat{j}$ around the closed path $OABC$ shown in the figure is:

$A$ particle of mass $m$ with initial kinetic energy $K$ approaches the origin from $x = +\infty$. Assume that a conservative force acts on it and its potential energy $V(x)$ is given by $V(x) = \frac{K}{\exp(3x/x_0) + \exp(-3x/x_0)}$, where $x_0 = 1 \ m$. The speed of the particle at $x = 0$ is

Given below are two statements: one is labelled as Assertion $A$ and the other is labelled as Reason $R$.
Assertion $A$: In a central force field,the work done is independent of the path chosen.
Reason $R$: Every force encountered in mechanics does not have an associated potential energy.
In the light of the above statements,choose the most appropriate answer from the options given below:

$A$ particle moves in a potential region given by the formula $U = K(x^2 + y^2 + z^2)$. Find the force acting on the particle.

The potential energy of a certain particle is given by $U = \frac{1}{2}\,(x^2 - z^2)$. The force on it is: