Derive an expression for torque in polar coordinate system ,with the help of appropriate figure.

Explanation

Suppose the position vector of the particle at point P (x,y) is `vecr` and it makes an angle `theta` with x -axis. Let the angle between the force vector `vecF` and position vector `vecr` be `phi`. If `vecF` an angle `alpha` with the positive direction of x-axis then
` phi=alpha -theta`
Now `F_(x)= Fcos alpha "and" F_(y) = F sin alpha`
If x and y are coordinates of the point P where PO = r and `angleXOP=theta `, then
x= rcos`theta ` and y = rsin`theta`
We have seen that , `tau= xF_(y) - yF_(x) " "cdots(1)`
Putting the various values in equation (1), we have
`tau= (rcostheta) (Fsinalpha)-(rsintheta) (Fcosalpha)`
`= rF(sinalphacostheta-cosalphasintheta)`
=rFsin`(alpha-theta)`
= r Fsin`phi ( because phi= alpha-theta)`
`:. tau= rFsin phi`
The above expression is the expression for torque in polar coordinates. Note that torque due to a `vecF` force depends upon the magnitude of force and displacement `vec(r)` of the force from the axis of rotation ( point O ).