One end of a light spring of spring constant $k$, is fixed to a wall and the other end is tied to a block placed on a smooth horizontal surface. In a displacement, the work done by the spring is $\frac{1}{2}k{x}^2$. The possible cases are

A force $F$ (power of $F$ is $p=$ constant) is applied on block $A$ of mass $m$ as shown in the figure, $F$ is parallel to the inclined plane. Then:

Two bodies of mass $m_1$ and $m_2 (m_2>m_1)$ are connected by a light inextensible string which passes through a smooth fixed pulley as shown. Then choose the correct option(s) 

The given plot shows the variation of $U$, the potential energy of interaction between two particles with the distance separating them is $r$. Then which of the following statements is/are correct:

A single conservative force $F\left(x\right)$ acts on a particle that moves along the $x$-axis. The graph of the potential energy with $x$ is given $x=5 \mathrm{m}$, the particle has a kinetic energy of $50 \mathrm{J}$ and its potential energy is related to the position $x$ as $U=15+{\left(x–3\right)}^2 \mathrm{Joule}$, where $x$ is in $\mathrm{meter}$.

In the following arrangement the pulley is assumed to be light and the string inextensible. The acceleration of the system can be determined by considering the conservation of a certain physical quantity. The physical quantity conserved and the acceleration, respectively, are

Assuming all the surfaces to be smooth, if the time period of motion of the ball is $N\times {10}^{–1} \mathrm{s}$, then find $N$. Neglect the small effect of the bend near the bottom $(g=10 \mathrm{m} {\mathrm{s}}^{-2})$.