surface * . In what follows we have taken the upward direction to be positive. Let us raise the ball up to a height h . The work done by the external agency against the gravitational force is mgh .
This work gets stored as potential energy. Gravitational potential energy of an object, as a function of the height h , is denoted by V ( h ) and it is the negative of work done by the gravitational force in raising the object to that hei g ht. V ( h ) = mgh If h is taken as a variable, it is easily seen that the gravitational force F equals the negative of the derivative of V ( h ) with respect to h . Thus, V(h) m g h The negative sign indicates that the gravitational force is downward.
When released, the ball comes down with an increasing speed. Just before it hits the ground, its speed is given by the kinematic relation, v = gh This equation can be written as m v = m g h which shows that the gravitational potential energy of the object at height h , when the object is released, manifests itself as kinetic energy of the object on reaching the ground. Physically, the notion of potential energy is applicable only to the class of forces where work done against the force gets ‘stored up’ as energy. When external constraints are removed, it manifests itself as kinetic energy.
Mathematically, (for simplicity, in one dimension) the potential * The variation of g with height is discussed in Chapter on Gravitation. energy V ( x ) is defined if the force F ( x ) can be written as ( ) F x This implies that F(x) x The work done by a conservative force such as gravity depends on the initial and final positions only. In the previous chapter we have worked on examples dealing with inclined planes. If an object of mass m is released from rest, from the top of a smooth (frictionless) inclined