A force $\vec{F} = (3x^2\hat{i} + 4y\hat{j})$ N acts on a particle as it moves from (0,0) to (1,1) m. Which of the following paths requires the least amount of work done by the force?

The potential energy as a function of the force between two atoms in a diatomic molecules is given by $U\left( x \right) = \frac{a}{{{x^{12}}}} - \frac{b}{{{x^6}}}$ , where a and b are positive constants and x is the distance between the atoms. The position of stable equilibrium for the system of the two atoms is given

A bullet of mass 10 g is fired horizontally with a velocity 1000 $m{s^{ - 1}}$ from a rifle situated at a height 50 m above the ground. If the bullet reaches the ground with a velocity 500 $m{s^{ - 1}}$,the work done against air resistance in the trajectory of the bullet is (g=$10m{s^{ - 2}})$

A box of mass 50 kg is pulled up on an incline 12 m long and 2 m high by a constant force of 100 N from rest. It acquires a velocity of $2{\rm{\;m}}{{\rm{s}}^{ - 1}}$ on reaching the top. Work done against friction $\left( {{\rm{g}} = 10{\rm{\;m}}{{\rm{s}}^{ - 2}}} \right)$ is

A particle of mass 'm' moves under the influence of a central force given by $F(r) = -\frac{k}{r^3}$, where 'k' is a positive constant. If the particle's initial velocity is 'v' at a distance 'r' from the center of the force, what is the minimum velocity required for the particle to escape to infinity?

Two bodies of mass M and m are moving with same kinetic energy. If they are stopped by same retarding force, then

A mass of M kg is suspended by a weightless string. The horizontal force that is required to displace it until the string makes an angle of ${45^0}$ with the initial vertical direction is

A body of mass 100 g is rotating in a circular path of radius r with constant velocity. The work done in one complete revolution is

A body of mass 6kg is under a force which cause displacement in it given by $S = \frac{{{t^2}}}{4}$ metres where t is time. The work done by the force in 2 seconds is

A 5 kg stone of relative density 3 is resting at the bed of a lake. It is lifted through a height of 5 m in the lake. If ${\rm{g}} = 10{\rm{m}}{{\rm{s}}^2}$ , then the work done is

The mass of ship is $2 imes {10^7}\;kg$. On applying a force of $25 imes {10^5}\;N$ , it is displaced through 25 m. After the displacement, the velocity acquired by the ship will be