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gravitational potential energy formula

What is potential energy?

Potential energy is defined as:“Energy due to position is called potential energy.”

When a diver jumps off a high  board into a swimming pool,he hits the water moving pretty fast,with a lot of kinetic energy.Where does that energy come from?The answer is that gravitational force (his weight) does work on the diver as he falls.The diver’s kinetic energy-energy associated with his motion-increases by an amount equal to work done.

However,there is a very useful alternative way to think about work and kinetic energy.This new approach is based on the concept of potential energy,which is energy associated with the position of a system rather than its motion.In this approach,there is gravitational potential energy even while the diver is standing on the high board.Energy is not added to the earth diver system as the diver falls,but rather a storehouse of energy is transformed from one form  (potential energy) to another (kinetic energy) as he falls.And also we will see how the work energy theorem explains this transformation.

If the diver bounces on the end of the board before he jumps,the bent board stores a second kind of potential energy called elastic potential energy.We will discuss elastic potential energy of simple systems such as a stretched or compressed spring.(An important third kind of potential energy is associated with the positions of electrically charged particles relative to each other.)

We will also prov e that in some cases the sum of a system’s kinetic and potential energy,called the total mechanical energy of the system,is  constant during the motion of the system.This will lead us to the general statement of the law of conservation of energy,one of the most fundamental and far reaching principles in all of science.

Gravitational potential energy:

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.

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.

Conservation of Mechanical Energy (Gravitational Forces Only):

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.

Watch also video about potential energy:

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