# Potential energy|Gravitational Potential Energy

## (6)Potential energy

• Potential energy is the energy stored in the body or a system by virtue of its position in field of force or by its configuration
• Force acting on a body or system can change its Potential energy
• Few examples of bodies possesing Potential energy are given below
i) Stretched or compressed coiled spring
ii) Water stored up at a height in the Dam possess PE
iii) Any object placed above the height H from the surface of the earth posses PE
• Potential energy is denoted by letter U
• In next two topics we would discuss following two example of PE
i)PE of a body due to gravity above the surface of earth
ii) PE of the spring when it is compressed orr elongated by the application of some external force

## (7) Gravitational Potential energy(near the surface of earth)

• All bodies fall towards the earth with a constant acceleration known as acceleration due to gravity
• Consider a body of mass m placed at height h above the surface of the earth
• Now the body begins to fall towards the surface of the earth and at any time t ,it is at height h' (h' < h) above the surface of the earth
• During the fall of the body towards the earth a constant force F=mg acts on the body where direction of force is towards the earth
• Workdone by the constant force of gravity is
W=F.d
W=mg(h-h')
W=mgh-mgh' -(12)
From equation (12) ,we can clearly see that work done depends on tbe difference in height or position
• So a potential energy or more accurately gravitational potential energy can be associated with the body such that
U=mgh
Where h is the height of the body from the refrence point
• if
Ui=mgh Uf=mgh' or
W=Ui-Uf=-ΔU           (13)
The potential energy is greater at height h and smaller at lower height h
• We can choose any position of the object and fix it zero gravitational PE level.PE at height above this level would be mgh
• The position of zero Potential energy is choosen according to the convenience of the problem and generally earth surface is choosen as position of zero PE
• One important point to note is that equation (13) is valid even for the case when object object of mass m is thrown vertically upwards from height h to h'( h' > h)
• Points to keep in mind
i) If the body of mass m is thrown upwards a height h above the zero refrence level then its PE increase by an amount mgh
ii) If the body of mass falls vertically downwards through a height h ,the PE of body decreases by the amount mgh

## (8) Conversion of Gravitational Potential Energy to Kinetic Energy

• Consider a object of mass placed at height H above the surface of the earth
• By virtue of its position the object possess PE equal to mgH
• When this objects falls it begins to accelerate towards earth surface with acceleration equal to acceleration due to gravity g
• When object accelerates it gathers Speed and hence gains Kinetic energy at the expense of gravitational PE
• To Analyze this conversion of PE in KE consider the figure given below

• Now at any given height h,PE of the object is given by the U=mgh and at this height speed of the object is given by v
• Change in KE between two height h1 and h2 would be equal to the workdone by the downward force
mv12/2 -mv22/2 =mgd           (14)
Where d is the distance between two heights .here one important thing to note is that both force and are in the same downwards direction
• Corresponding change in Potential energy
U1-U2=mg(h1-h2)=-mgd           (15)
Here the negative sign indicates the decrease in PE whch is exacty equal to the gain in KE
• From equation (14) and (15)
mv12/2 -mv22/2=U2-U1
or KE1 + PE1=KE2 + PE2
• Thus total energy of the system KE and gravitational PE is conserved .There is a conversion of one form of energy into another during the motion but sum remains same
• This can be proved as follows
At height H,velocity of the object is zero
ie
at H=H ,v=0 => KE=0
So $KE+PE=mgH$

at H=h ,v=v => KE=mv2/2=mg(H-h) and U=mgh
So KE+PE=mgH

at H=0 ,v=vmax => KE=mvmax2/2=mgH
So KE+PE=mgH
• This shows that total energy of the system is always conserved becuase total energy KE+PE=mgH ,which remains same as plotted in the figure given below

• We know conclude that energy can neither be produced nor be destroyed .It can only be transformed from one form to another