In the figure, the variation of the potential energy of a particle of mass $m=2 \mathrm{kg}$ is represented with respect to its $x$-coordinate. The particle moves under the effect of this conservative force along the$x$-axis. Which of the following statements are correct about the particle?

A block of mass $m$ is pushed against a spring of spring constant $k$ fixed at one end to a wall. The block can slide on a frictionless table as shown in the figure. The natural length of the spring is $L_0$ and it is compressed to one-fourth of natural length and the block is released. Find its velocity as a function of its distance $\left(x\right)$ from the wall and maximum velocity of the block. The block is not attached to the spring.

Consider a one-dimensional motion of a particle with total energy, $E$. There are four regions $A,B,C$ and $D$in which the relation between potential energy $U$, kinetic energy $K$ and total energy $E$ is as given below.

Region $A: U> E$

Region $B:U<E$

Region $C: K>E$

Region $D:U>\text{ K}$

State with reason in each case whether a particle can be found in the given region or not.

A curved surface is shown in the figure. The portion $BCD$ is free of friction. There are three spherical balls of identical radii and masses. Balls are released from rest one by one from $A$which is at a slightly greater height than $C$.

With the surface $AB$, ball $1$ has large enough friction to cause rolling down without slipping; ball $2$ has a small friction and ball $3$ has a negligible friction.

(a) For which balls is total mechanical energy conserved?

(b) Which ball(s) can reach $D$?

(c) For balls which do not reach $D$, which of the balls can reach back $A$?

A particle is moved from $\left(0,0\right)$ to $\left(a,a\right)$ under a force $F=\left(3\hat{\mathrm{i}}+4\hat{\mathrm{j}}\right)$ from two paths. Path $1$ is $OP$ and path $2$ is $OQP$.  Let $W_1$ and $W_2$  be the work done by this force in these two paths. Then,

A ball is suspended from the top of a cart by a string of length $1.0\mathrm{ }\mathrm{m}$. The cart and the ball are initially moving to the right at constant speed $V$ as shown in figure $\left(1\right)$. The cart comes to rest after colliding and sticking to a fixed bumper, as in figure $\left(2\right)$. The suspended ball swings through a maximum angle $60°.$ The initial speed $V$ is $($take $g=10 \mathrm{m} {\mathrm{s}}^{-2})$,