# Why should angular momentum be conserved in this case?

Consider two disks (not friction-less) with some moment of inertia ($$I_1$$ and $$I_2$$). Both of them are given angular velocities ($$\omega_1$$ and $$\omega_2$$) both in same sense.

Now if we bring both disks in contact after some time they will have common angular velocity. Now my text says that the new angular velocity ($$\omega$$) is given by the equation $$I_1\omega_1+I_2\omega_2=I\omega$$

But how can angular momentum be conserved in this case? Isn't friction applying torque?

And if the explanation contains that friction is applying internal torque then please explain.

• Would be good to mention that general form of conservation of angular momentum looks a bit different : $$\sum_i L_i = \sum_i I_i~\omega_i = \text{const}$$ So in your 2 disk case, general form of angular moment conservation will be : $$I_1~\omega_1 + I_2~\omega_2 = I_1~\omega^\prime_1 + I_2~\omega^\prime_2$$ How your book text deduced common moment of inertia and common angular velocity - that's a different and (to me) unknown story. Jul 1 '20 at 18:56

Torque here is not external, you can tell because the total angular momentum in the system is the sum of the angular momentum of the two disks. Therefore the two disks are what makes up the system, neither of them are an external object. They only exchange momentum between each other, as they have both applied torques to each other.

It is the same concept as linear momentum, if you have a system of two pool balls and they collide, they apply forces to each other and exchange momentum, but unless there is an outside object that's taking momentum from them (which happens when something applies an external force) the total momentum is conserved.

So unless you bring in air friction, put brakes on the disks to remove energy as heat, bring in a third disk that has a magnet attached to remove energy as an induced current, etc. there is no external force.

If the system is the two discs then the frictional forces apply internal torques which have a net value of zero - the internal torques are opposite in direction and equal in magnitude.
If no external torques are applied then angular momentum is conserved.

The law of conservation of angular momentum states that when no external torque acts on an object, no change of angular momentum will occur.

Yes there is friction between the discs,when they come into contact .

Consider the resultant of the friction forces acting on the discs to be F. As shown above they are an action-reaction pair.They are internal forces. One wouldn't be there if not for the other. So if you consider the torques due to these forces, they cancel-out as they will be opposite and equal to each other.

Hence we can safely apply the Law of conservation Of Angular Momentum.

• what if the radius of both the disk are different, then I think the resultant torque wouldn't cancel each other?, but yes I agree that forces on both the disk would be equal and opposite. Jun 29 '20 at 12:45
• Where the surfaces come in contact, the forces are equal and opposite. If one disk is larger, it will only come in contact where the smaller disk touches it. This describes torque for a different problem, but it might give you some insight about how you can get torques from the individual forces. Also note you can consider this to be a single system where the friction forces are internal, or two systems where friction from one is an external force on the other. In the second case, the two systems exert equal and opposite torques on each other. Jun 29 '20 at 13:30

For a system of bodies not subject to external forces the conservation of linear and angular momentum are incontestable theorems of Newtonian mechanics. but mechanical energy isn't usually conserved.

Indeed in our system from the conservation of angular momentum

$$I_1 \omega_1 + I_2 \omega_2 = (I_1+I_2) \omega$$

it easily follows that

$$\frac{1}{2} I_1 \omega_1^2 + \frac{1}{2} I_2 \omega_2^2 \geq \frac{1}{2} (I_1+I_2) \omega^2$$

The dissipation of mechanical energy due to friction will cause the discs to heat up.