1)

The rotational first law states that a body in rotation or at rest tends to keep its state of motion as long as there is no external torque applied to it. The mathematical expression refers to the conservation of the angular momentum:

**L** = **r **x **p** = ** r **x (m**v) = **m*(**r** x **v**)

where x is the vectorial product and bold letters represent vector quantities.

2)

We know that the torque applied is

`T = I*epsilon`` `**` ` **

where `epsilon ` is the angular acceleration, thus

`epsilon = omega xx (omega xx r) = omega xx v`

Now we can state the relation between applied torque and angular speed since momentum of inertia ` I` is a scalar quantity. When a variable (in direction and modulus) external torque is applied the axis of angular speed `omega ` (which is perpendicular to the torque) starts to change direction to remain perpendicular to the instantaneous torque and also its magnitude tend to follow the variations of the torque applied.

The relation between external torque and angular momentum is

`T =I*epsilon =I*(d(omega)/dt) = (d(I*omega))/(dt) =(dL)/(dt)`

3) and 4)

The rotational second law is

`T= I*epsilon `

which in words can be expressed as: the applied external torque is equal to the moment of inertia of the body multiplied by its angular acceleration.