Home » Mechanics » gravitational potential energy formula derivation and examples

# gravitational potential energy formula derivation and examples

## Gravitational potential energy definition

We know that a particle gains or loses kinetic energy because it interacts with other objects that exert forces on it.During any interaction,the change in a particle’s kinetic energy is equal to the work done on the particle by the forces that act on it.

## Examples of gravitational potential energy in everyday life

In many situations it seems as through energy has been stored in a system ,to be recovered later.For example,you must do work to lift a heavy stone over your head.It seems reasonable that in hosting the stone into kinetic energy when you let the stone fall.

This example points to the idea of an energy associated with the position bodies in a system.This kind of energy is a measure of the potential or possibility for work to be done;when a stone is raised into air,there is a potential for work to be done on it by the gravitational force,but only if the stone is allowed to fall to the ground.For this reason,energy associated with position is called potential energy.Our discussion suggests that there is potential energy associated with a body’s weight and its hight above the ground.We call this gravitational potential energy.

We now have two ways to describe what happens when a body falls without air resistance. One way is to say that gravitational potential energy decreases and the falling body’s kinetic energy increases. The other way,which we learned,is that a falling body’s kinetic energy increases because the force of the earth’s gravity (the body’s weight) does work on the body. Later in the section we’ll use the work-energy theorem to show that these two descriptions are equivalent.

## Derivation of gravitational potential energy

Let a body of mass m as shown in figure is raised up through height h from the ground.The body will acquire potential energy equal to the work done is lifting it to height h.

So potential energy P.E = (F)(h)

P.E=(W)(h) …….(1)

Where  weight of the body is W= mg

By putting value of W in equation (1) we have:

P.E =mgh

## Conservation of Mechanical Energy

To see what gravitational potential energy is good for, suppose the body’s weight is the only force acting on it,so:

The body is then falling freely with no air resistance,and can be moving either up or down. Let is speed at point y1 be v1 and let its speed at y2 be v2. The work energy-theorem,say that the total work done on the body equals the change in the body’s kinetic energy:

Wgrav = ∆K = K2 – K1 .

If gravity is the only force that acts,then from equation,

Wnot = Wgrav = -∆Ugrav = Ugrav,1 – Ugrav,2.

Putting these together,we get:

∆K = -∆Ugrav    or K2 – K1 = Ugrav,1 – Ugrav,2

which we write as:

K1 + Ugrav,1  = K2 + Ugrav,2  ……….eq (1)

Or

mv + mgy1  ………….eq (2)

The sum K+ Ugrav of kinetic and potential energy is called E,the total mechanical of the system. By “system” we mean the body of mass m and the earth considered together, because gravitational potential energy U is a shared property of both bodies. Then E1 = K1 + Ugrav,1 is the total mechanical energy at y1 and E2 = K2 + Ugrav,2 is the total mechanical energy at y2 . Equation (1) says that when the body’s weight is the only force doing work on it E1 = E2 . That is, E is constant;it has the same value at y1 and y2 . but since the positions y1 and y2 are arbitrary points in the motion of the body,the total mechanical energy E has the same value at all points during the motion:

E = K + Ugrav  = constant

A quantity that always has the same value is called a conserved quantity. When only the force of gravity does work,the total mechanical energy is constant-that is, is conserved. This is our first example of the conservation of mechanical energy.

When a throw a ball into the air, its speed decreases on the way up as kinetic energy is converted to potential energy; ∆K < 0 and ∆ Ugrav >0. On the way back down,potential energy is converted back to kinetic energy and the ball’s speed increases; ∆K > 0 and ∆ Ugrav < 0. But the total mechanical energy (kinetic plus potential) is the same at every point in the motion,provided that no force other than gravity does work on the ball (that is,air resistance must). While this athlete is in midair,only gravity does work on him(if we neglect the minor effect of air resistance). Mechanical energy E-the sum of kinetic and gravitational potential energy-is conserved.

It’s still true that the gravitational force does work on the body as it moves up or down,but we no longer have to calculate work directly;keeping track of changes in the value of Ugrav takes care of this completely.