# Angular Momentum and assymetric axis

The question I came across ,

If a semicircular disc rotates uniformly (const. angular velocity) about an axis passing through its Centre of mass , and prependicular to its plane , do we need an external torque to sustain its motion

My attempt includes going to the original definition of torque as rate of change of angular momentum , which I believe is constant here. The book mentions something of an assymetric axis , and thus not a constant angular momentum.

As per more reading I believe it has something to do with conditions where angular velocity and momentum of a rigid body are not parallel.

One condition that I can feel is when the axis is not fixed but how does assymetric axis fulfills this condition as in the question above.

The angular velocity of a rigid body will stay constant without any torque applied if the angular velocity is parallel to a principal axis. You can see that directly from Euler equations

$$I_i \frac{d \omega _i}{dt} + (I_k - I_j)\omega _j \omega _k= N_i$$

If your angular velocity is completely oriented on a single principal axis at=0 and the torque is zero, then we have that $$\omega _j \omega _ k$$ is 0 at t=0 and so the ODE's reduce to $$\frac{d \omega _i}{dt} =0$$ at t=0.

Which gives that the three components of the angular velocity do not instantaneously change at t=0. Hence the two components of the angular velocity that are zero stay zero at t=0. It is then easy to see by iteration that this is actually true for all t, i.e. the angular velocity stays constant, so the object sustain its original motion.

So as for your question, ask yourself if the axis perpendicular to the semicircular disk is a principal axis.

Also, as a side note, the angular momentum is always constant if there is no net total torque.

• I have added my books answer as an edit , please refer to it , and bring out the flaw in my reasoning Commented Apr 1, 2019 at 4:50
• Well, maybe I am wrong, but I don't agree with your book. You should be able to have a constant motion without any torque around the axis going into the CM perpendicular to the plane of the object, whether it is a disk or a wheel. A pancake object in the x-y plane always sustain constant motion around the z-axis through its CM. Commented Apr 1, 2019 at 15:43