# Work Energy Theorem and Potential Energy

This theorem is a very important tool that relates the work to kinetic energy. According to this theorem, the work done by all forces (conservative or nonconservative, external or internal) acting on a particle or an object is equal to the change in kinetic energy of it.

\(W_{net}=\Delta KE=k_{f}-K_{i}\)

Let, \( \overrightarrow {F_{1}},\overrightarrow {F_{2}}, – – – – \) be the individual forces acting on a particle. the resultant force is

\( \overrightarrow {F}=\overrightarrow {F_{1}},\overrightarrow {F_{2}}, – – – – \)

###### and the work done by the resultant force is

\(\begin{aligned}W=\int F\cdot d\overrightarrow {r}=\int \left( \overrightarrow {F_{1}}+\overrightarrow {F_{2}}+\ldots \right) \cdot d\overrightarrow {r}\\ W=\int \overrightarrow {F_{1}}\cdot d\overrightarrow {r}+\int \overrightarrow {F_{2}}\cdot d\overrightarrow {r}+\ldots \end{aligned}\)

Where, \(\int \overrightarrow {F_{1}}\cdot dr \) is the work done on the particle by \(\overrightarrow {F_{1}}\) and so on.

###### Thus work-energy theorem can also be written as work done by all the resultant forces which are also equal to the sum of the work done by the individual forces is equal to change in kinetic energy.

**Regarding the work-energy theorem, these points are important:-**

- If Wnet is positive then \(k_{f}-k_{i}=positive\)
- This theorem can be applied to the non-inertial frame also. in a non-inertial frame it can be written as work done by all the forces including the pseudo force is equal to change in kinetic energy in the non-inertial frame.

**Potential energy:-**

The energy possessed by a body or system by virtue of its position or configuration is known as the potential energy.

For example,

A block attached to a compressed or elongate spring possessed some energy called elastic potential energy. this block has the capacity to do work. similarly, a stone when released from a certain height also has energy in the form of gravitational potential energy. two charged particle kept at a certain distance has electrical potential energy.

**Regarding the potential energy, it is important to know that**

###### It is defined for a conservative force field only for non-conservative force field it has no meaning the change in potential energy of a system corresponding to a conservative force field is given by

\(dU=-\overrightarrow {F}\cdot d\overrightarrow {r}=-dW\)

We generally choose the reference point at Infinity and assume potential energy to be zero there.

\(U=-\int ^{r}_{\infty }\overrightarrow {F}\cdot d\overrightarrow {r}=-W\)

**“The potential energy of a body or system is negative of work done by the conservative force field is bringing it from infinity to the present position”**

**Regarding the potential energy, these points are important**

- Potential energy can be defined only for the conservative force field and it should be considered to be a property of the entire system rather than assigning it to any specific particle.
- Potential energy depends on the frame of reference.

**These are following three types of potential energy which are generally used in Physics**

**Elastic potential energy:-**

\(U=\dfrac {1}{2}kx^{2}\)

Elastic potential energy is always positive

**Gravitational potential energy:-**

The gravitational potential energy for two particles of mass m1 and m2 separated by a distance r is given by

**Electric potential energy:-**

The elastic potential energy of two point charges Q1 and Q2 separated by a distance R in a vacuum is given by

\(U=\dfrac {1}{4\pi \xi _{0}}\cdot \dfrac {q_{1}q_{2}}{r}\)

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