If $\omega = \frac{v}{r}$, why do we need torque?

Question

If $\omega = \frac{v}{r}$, then why do we need torque and angular acceleration? The velocity v can be found just by Newton's second law of motion $F = ma => a = F/m$ and $v = v_0 + at$ . Then we could find the angular velocity by dividing the linear velocity with a radius.

Answer 1

If you have a single particle then you can indeed easily describe its motion using Newton's laws. However in rotating systems we are often dealing with continuous bodies not point particles. These are characterised by a moment of inertia, rather than just by a mass, and we have an equation analogous to Newton's second law:

$$ T = I\dot{\omega} = I\ddot{\theta}$$

that corresponds to:

$$ F = ma $$

This makes torque and angular acceleration very useful concepts for dealing with real world systems. You could describe your object as a integral of infinitesimal volume elements, but that would turn all the equations of motion into integral equations and make life unnecessarily complicated.