Potential energy near the surface of the Earth
The gravitational potential energy (U) at some height h is equal to the amount of work required to take the object from ground to that height h with constant velocity.
Let us consider a body of mass m being moved from ground to the height h against the gravitational force as shown in Figure 4.8.
The gravitational force g acting on the body is, g = - mg jˆ (as the force is in y direction, unit vector ˆj is used). Here, negative sign implies that the force is acting vertically downwards. In order to move the body without acceleration (or with constant velocity), an external applied force a equal in magnitude but opposite to that of gravitational force g has to be applied on the body i.e., a = - g . This implies that a = + mg jˆ. The positive sign implies that the applied force is in vertically upward direction. Hence, when the body is lifted up its velocity remains unchanged and thus its kinetic energy also remains constant.
The gravitational potential energy (U) at some height h is equal to the amount of work required to take the object from the ground to that height h.
Since the displacement and the applied force are in the same upward direction, the angle between them, θ=0o. Hence, cos00 =1 and |a| = mg and |d| = dr .
Note that the potential energy stored in the object is defined through work done by the external force which is positive. Physically this implies that the agency which is applying the external force is transferring the energy to the object which is then stored as potential energy. If the object is allowed to fall from a height h then the stored potential energy is converted into kinetic energy.