A time dependent force $\mathrm{F}=6\mathrm{t}$ acts on a particle of mass $1 \mathrm{kg}$. If the particle starts from rest, the work done by the force during the first $1 \mathrm{s}$ will be:

Two identical cylindrical vessels with their bases at the same level each contain a liquid of density $\rho$ as shown in the figure. The height of the liquid in one vessel is $h_2$ and in other vessels $h_1$, the area of either base is $A$. Find the work done by gravity in equalizing the levels when the two vessels are connected.

A uniform rope of linear mass density $\mathrm{\lambda }$ and length $l$ is coiled on a smooth horizontal surface. One end is pulled up with constant velocity $\mathrm{v}.$ Then find average power applied by the external agent in pulling the entire rope just off the ground ?

A uniform rope of linear mass density $\lambda$ and length $l$ is coiled on a smooth horizontal surface. One end is pulled up with a constant velocity $v$. The maximum power delivered by the agent in pulling up the rope is?

Two blocks $A \& B$ of mass $m \& 2m$ respectively are joined to the ends of an undeformed massless spring of spring constant $k$. They can move on a horizontal smooth surface. Initially $A \& B$ have velocities $u$ towards left and $2u$ towards right respectively. Constant forces of magnitudes $F \text{and} 2F$ are always acting on $A \text{and} B$ respectively in the directions shown. Find the maximum extension in the spring during the motion.

Figure shows a rod of length $L$ which is uniformly charged with linear charge density $\lambda$ kept on a smooth horizontal surface. Right end of rod is in contact with a vertical fixed wall. A block of mass m and charge q is projected with a velocity $v$ from a point very far from rod in the line of rod. Find the distance of the closest approach between the block & the left end $A$ of the rod.

A non-conducting disc of radius $a$ and uniform positive surface charge density $\sigma$ is placed on the ground with its axis vertical. A particle of mass $m$ and positive charge $q$ is dropped along the axis of the disc from a height $H$ with zero initial velocity. The particle has, $\frac{q}{m}=\frac{4{\epsilon }_{0}g}{ \sigma }$.

(i) Find the value of $H$ if the particle just reaches the disc.

(ii) Sketch the potential energy of the particle as a function of its height and find its equilibrium position.