## Work, Energy and Power

# Potential energy and Law of Conservation of Energy

- Potential energy of a mass ‘m’ at height ‘h’ u = mgh.
- Potential energy of a stretched spring \tt u = \frac{1}{2} kx^{2}.
- Zero potential point is that where gravitational potential = 0.
- In a conservative force work done in round trip is zero.
- In a Non-conservative force work done in round trip is non zero.
- Relation between force and potential energy \tt F = - \frac{du}{dx}
- In stable equilibrium \tt F = 0 = \frac{du}{dx}
- In stable equilibrium potential energy is minimum.
- In stable equilibrium when a particle is slightly displaced it comes to initial position.
- In unstable equilibrium, when a particle is slightly displaced it moves further away.
- In unstable equilibrium potential energy is maximum.
- In neutral equilibrium when a particle is displaced it remains in that position.
- Potential energy is constant in Neutral equilibrium.
- In neutral equilibrium force is zero.
- In neutral equilibrium \tt \frac{d^{2}u}{dx^{2}} = 0
- For a particle moving in circular path centripetal force = \tt \frac{mv^{2}}{r} = \frac{K}{r}
- Kinetic energy of a particle in circular path = \tt \frac{K}{2r}
- Potential energy of a particle in circular path = \tt \frac{-K}{r}
- Total energy of a particle in circular path = \tt \frac{-K}{2r}
- One form of energy may be converted into other form called conservation of energy.
- In the principle of conservation of energy we include mass into energy.
- The principle of conservation of energy cannot be proved mathematically but is an empirical principle.
- Einstein’s mass energy equivalence E = Δmc
^{2}. - The sum of kinetic energy and potential energy is constant.
- W
_{con}+ W_{non con }+ W_{ext}= KE_{2}− KE_{1}

### Potential Energy View the Topic in this video From 27:34 To 58;17

### View the Topic in this video From 13:46 To 58:46

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1. Potential energy = mgh.

2. Potential energy of the spring = \tt \frac{1}{2}kx^{2}

3. The electric potential energy of two point charges q_{1} and q_{2} separated by a distance r in vacuum is given by \tt U=\frac{1}{4\pi\varepsilon_{0}}\cdot\frac{q_{1}q_{2}}{r}

4. If spring is stretched from initial position x_{1} to final position x_{2} then work done = Increment in elastic potential energy

=\frac{1}{2}k\left(x_2^2-x_1^2\right)

5. For two particles of masses m_{1} and m_{2} separated by a distance r

Gravitational potential energy \tt U=-\frac{G\ m_{1}m_{2}}{r}

6. If a body of mass m is at height h relative to surface of earth then

Gravitational potential energy \tt U=\frac{mgh}{1+\frac{h}{R}}

7. If point mass m is pulled through a height h then work done W = mgh

8. Work done in pulling the hanging portion on the table.

\tt W=\frac{MgL}{2n^{2}}

9. Work done to raise the centre of mass of the chain on the table is given by

\tt W=\frac{MgL}{2n^{2}}

10. It is defined as the sum of kinetic energy K and potential energy U. i.e. E = K + U

11. Principle of conservation of mechanical energy: K + U = constant.