A particle is subjected to a conservative force whose potential energy is given by $U(x) = ax^2 - bx^3$, where 'a' and 'b' are positive constants. At what value of 'x' is the particle in stable equilibrium?

A man throws a piece of stone to a height of 12 m where it reaches with a speed of $12\;{\rm{m}}{{\rm{s}}^{ - 1}}$. If he throws the same stone such that it just reaches this height, the percentage of energy saved is nearly

A spring with spring constant 'k' is compressed by a distance 'x'. The potential energy stored in the spring is:

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

A vertical spring with force constant k is fixed on a table .A ball of mass m at a height h above the free upper end of the spring falls vertically on the spring, so that the spring is compressed by a distance d. The net work done in the process is

If a body of mass 200 g falls from a height 200 m and its total P.E. is converted into K.E. at the point of contact of the body with earth surface, then what is the decrease in P.E. of the body at the contact ($g = 10\;m/{s^2}$)

A 2.0 kg block is dropped from a height of 40 cm onto a spring of spring constant $k = 1960\;{\rm{N}}{{\rm{m}}^{ - 1}}$ . Find the maximum distance the spring is compressed

The potential energy of a 1 kg particle free to move along the x-axis is given by $V\left( x \right) = \left( {\frac{{{x^4}}}{4} - \frac{{{x^2}}}{2}} \right){\rm{J}}$.The total mechanical energy of particle is 2 J. Then, the maximum speed $\left( {{\rm{in\;m}}{{\rm{s}}^{ - 1}}} \right)$ is

The bodies of masses 1 kg and 5 kg are dropped gently from the top of a tower. At a point 20 cm from the ground, both the bodies will have the same

A block of mass 'm' slides down a frictionless curved track from a height 'h'. It then collides elastically with a stationary block of mass '2m'. After the collision, the smaller block momentarily comes to rest. What is the maximum height the larger block will reach?

A small ball attached to a massless string is undergoing vertical circular motion. At the highest point of its trajectory, the tension in the string is zero. Which of the following statements is TRUE about the ball's mechanical energy as it moves from the highest point to the lowest point in the circle?