- Introduction
- Work
- Work done by variable force
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- Mechanical Energy
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- Kinetic energy
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- Potential energy
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- Gravitational PE near the surface of the earth
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- Conversion of Gravitational PE to KE
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- PE of the spring
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- Conservation of energy of spring mass system
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- Conservative Forces
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- Power
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- Principle of conservation of energy
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- Solved examples

- 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 PE

- Few examples of bodies possesing PE 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

- 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 workdone 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

U_{i}=mgh U_{f}=mgh^{'}or

W=U_{i}-U_{f}=-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 PE 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

- 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 h
_{1}and h_{2}would be equal to the workdone by the downward force

mv_{1}^{2}/2 -mv_{2}^{2}/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

U_{1}-U_{2}=mg(h_{1}-h_{2})=-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)

mv_{1}^{2}/2 -mv_{2}^{2}/2=U_{2}-U_{1}

or KE_{1}+ PE_{1}=KE_{2}+ PE_{2} - Thus totat 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=mv^{2}/2=mg(H-h) and U=mgh

So KE+PE==mgH at H=0 ,v=v_{max}=> KE=mv_{max}^{2}/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

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