11.12:
Rotation of Asymmetric Top
A spherically symmetric object rotating about any diameter has a constant moment of inertia.
If the object has no specific symmetry, it can be rotated about three axes. The moment of inertia about these three axes of rotation is different. Such an object is called an asymmetric top.
For example, consider a table tennis racket that can be rotated about three rotational axes. The moment of inertia about these three rotational axes measures how mass is distributed around each.
It is easier to rotate the table tennis racket about the axis of rotation for which the moment of inertia is the smallest, which means that for a given amount of torque, the racket will acquire maximum angular velocity about this axis of rotation.
For the axis about which the moment of inertia is maximum, the racket rotates slowly.
The racket's rotation is stable about these two axes of rotation, but becomes unstable when rotated about the intermediate axis of rotation.
11.12:
Rotation of Asymmetric Top
By definition, a spherically symmetric body has the same moment of inertia about any axis passing through its center of mass. This situation changes if there is no spherical symmetry. Since most rigid bodies are not spherically symmetric, these require special treatment.
The relationship between the angular momentum of any rigid body and its angular velocity, both of which are vectors, involves the moment of inertia. The moment of inertia is a scalar quantity only for spherically symmetric rigid bodies. Otherwise, the moment of inertia is not a scalar quantity and is called a tensor. Scalars and vectors are special cases of tensors.
To relate angular momentum and velocity vectors, six independent values are required to describe the moments of inertia along the three orthogonal axes in 3D space. In special cases, such as when unique independent axes of rotation are chosen, only three numbers are sufficient to describe moment of inertia. These are called the principal axes of rotation and principal moments of inertia, respectively.
An object with three unequal moments of inertia is called an asymmetric top. The mathematics of its rotation is complicated, but it can be simplified by considering conservation principles. The angular momentum vector is constant if there is no external torque. Its magnitude is also conserved. This condition provides one constraint on the angular speeds. The other constraint is that the total kinetic energy is also conserved.
Suggested Reading
- OpenStax. (2019). University Physics Vol. 1. [Web version]. Retrieved from https://openstax.org/books/university-physics-volume-1/pages/10-4-moment-of-inertia-and-rotational-kinetic-energy
- Retrieved from https://www.maths.ed.ac.uk/~v1ranick/sylv/inertia.pdf
- Retrieved from Libretext https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_%28Tatum%29/04%3A_Rigid_Body_Rotation/4.06%3A_Force-free_Motion_of_a_Rigid_Asymmetric_Top