Conservative and Non conservative forces

What are Conservative and Non conservative forces?

If the work done on particle by a force is independent of how particle moves and depends only on initial and final position of the objects then such a force is called conservative force . Similarly if the word done is not independent of the path then that force is non-conservative force

This can also be states as if the work done by the force in closed loop is zero then it is a conservative force ,otherwise it is a non-conservative force.Non-conservative forces are called dissipative force

For conservative Force

$W_{closed \; loop} =\oint \mathbf{F}.d\mathbf{r} =0$

For non-conservative Force

$W_{closed \; loop} =\oint \mathbf{F}.d\mathbf{r}  \neq  0$

Examples of Conservative and Non conservative forces

Here is the list of Conservative and Non-conservative forces

Conservative Forces	Non conservative Forces
Gravitation $(mg)$	Friction$(\mu N)$
Elastic Forces $(kx)$	Viscosity
Electric Forces $(\frac{kq}{r^{2}})$	Air resistance
	Motor or rocket propulsion
	The tension in the cord $(T)$

Some Points about Conservative forces and Non-conservative forces

1) Potential energy is defined for Conservative forces

2) An object that starts at a given point and returns to the same point, then network done by the conservative force is zero while in case of non-conservative force, it will not be zero

3) When several conservative forces act on a particle, the potential energy is the sum of the potential energies for each force

4) Potential energy belongs to the system not to single particle as it is associated with Force and force on one object is always exerted by other particles

5) If only conservative forces are acting, the mechanical energy(K.E+P.E) of the system remains constant

How to find if the force is conservative

we can use two ways

1. 1. we can calculate the work done between two points across two paths and compare if the work done is same.If it is same, it is conservative force
   2. We can use below equation
      $\frac{\partial F_x}{\partial y}=\frac{\partial F_y}{\partial x}$

Example-1

A force $\mathbf{F}=xy \mathbf{i} + xy\mathbf{j}$ acts on a particle moving a XY plan.Find out if the force is conservative

Solution

$\frac{\partial F_x}{\partial y} = x$

$\frac{\partial F_y}{\partial x} = y$

Now $\frac{\partial F_x}{\partial y} \neq \frac{\partial F_y}{\partial x} $, So it is not conservative force

Example -2

Find out if the below force is conservative

$\mathbf{F}=xy^3 \mathbf{i} + x^3y\mathbf{j}$

Solution

$\frac{\partial F_x}{\partial y} = 3xy^2$

$\frac{\partial F_y}{\partial x} = 3yx^2$

Now $\frac{\partial F_x}{\partial y} \neq \frac{\partial F_y}{\partial x} $, So it is not conservative force

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