Dear students in 9th and 10th standard we have learnt about the energy, the forms of energy. We categorised the mechanical energy in to following forms:
- Kinetic energy,
Kinetic energy is the energy that possesses by the body due to its motion. If body of mass ‘m’ moves with the velocity of ‘v’ then kinetic energy of body is given as,
Kinetic energy = 1/2 mv2
- Potential energy.
Potential energy is the energy that body possesses due to its shape/size/mass etc.
The potential energy of an object at height ‘h’ above the surface of earth is given as,
∴ Potential energy = mgh
Let’s learn the Conservation of mechanical energy …….!
Statement: Energy can neither be created nor be destroyed. It can be transferred from on form to another. i.e. the energy is always inter convertible.
According to the above statement, the sum of kinetic energy and potential energy of mechanical system is always constant.
∆KE +∆PE = 0
Or,
Kinetic energy + Potential Energy = constant
Hence the alternating statement for conservation of energy can be given as; the total mechanical energy of a system is conserved if the forces, doing work on it, are conservative.
Let’s prove the conservation of mechanical energy….!
Imagine children enjoying the trip of giant wheel. Consider the following position of a boy/girl as shown in diagram,
Let, H= higher position at height ‘h’ from ground
M= middle position at height ‘x’ above ground
L= lower position on ground.
u=initial velocity of children at higher position
v= final velocity of children at lower position
g= acceleration due to gravity.
Let ‘m’ is mass of boy which is moving from higher to lower position in time‘t’.
Case:1) When boy is at higher position and reaching towards ground, kinetic energy increase due to gravity at the same instant potential energy goes on decreases. In this case potential energy is given as,
∴ Potential energy = m × g × h …….(1)
Now kinetic energy will be,
∴ Kinetic energy of body= 1/2 mu2
∴ Kinetic energy of body = 0
Total energy of boy will be sum of kinetic energy and potential energy,
∴Total energy = Kinetic energy + potential energy
∴ Total energy = 0+ mgh
∴ Total energy = mgh………………………(A)
Case:2) When boy is at lower position: kinetic energy is maximum and potential energy is minimum and close to zero.
∴ Potential energy = 0…………….(3)
Now kinetic energy will be,
∴ Kinetic energy of body = 1/2 mv2
Now from 3rd equation of motion we have,
∴ 2as =v2 – u2
∴ 2gh =v2 – 0
∴ v2 = 2gh
∴ Kinetic energy of body = 1/2 m × 2gh
∴ Kinetic energy of body = mgh ………….(4)
Total energy of boy will be sum of kinetic energy and potential energy,
∴Total energy = Kinetic energy + potential energy
∴Total energy = mgh + 0
∴ Total energy = mgh………………………(B)
Case:3) When boy is at midway position: At midway position, kinetic energy and potential energy both will present then,
∴ Potential energy = mg(h – x)
∴ Potential energy = mgh – mgx …………….(5)
Now kinetic energy will be,
∴ Kinetic energy of body = 1/2 mv2
Now from 3rd equation of motion we have,
∴ 2as =v2 – u2
∴2gx = v2 – 0
∴ v2 = 2gx
∴ Kinetic energy of body = 1/2 m × 2g(h – x)
∴ Kinetic energy of body = mgx ………….(6)
Total energy of boy will be sum of kinetic energy and potential energy,
∴Total energy = Kinetic energy + potential energy
∴ Total energy = mgx + mgh – mgx
∴ Total energy = mgh………………………(C)
From equations (A), (B) and (C) we can say that the total energy at all above points is constant. i.e. constant or conserved.