# What happens to moment of inertia if a body is divided into 2 congruent part?

If a circular disk is cut in half, and a square plate is divided into 2 identical right angled triangle, when the axis of rotation is not changed i.e if the square plate was rotating about an axis passing through it's center perpendicular to the plane of the object, how will 2 right angled triangular plates rotate about the same axis?

Moments of inertia stay the same when an object is divided into parts, congruent or not, as long as the parts are not moved apart. This is because moments of inertia depend on how the object’s mass is distributed in space, not on how its mass is divided up into “parts”.

When you compute moments of inertia, you integrate over infinitesimal mass elements $$dm$$. It doesn’t matter which “part” $$dm$$ is in. It only matters how far it is from the axis of rotation.

• The objects can of course be slid up or down the axis of rotation, or even rotated a fixed angle around the axis.
– cms
Mar 9 '19 at 12:43

You need to integrate it out by pieces, but my guess was to reduce.

If changing the axis of rotation to the center of newly created pieces:(https://en.wikipedia.org/wiki/List_of_moments_of_inertia) it reduces, by $$1/2$$ because $$L^2$$ dependence for a rigid stick rotate at the mid point in the plane.

If not changing the axis of rotation: it actually increased the initial by twice.

Image you are rotating a stick, if it's in one piece, it's easy for you to rotate. If you cut them in two and held them together at the very edge and then rotates, won't you feel more "heavy"? If it breaks, you need much more angular momentum to maintain the motion, or it would fly away.

• If the factor were smaller (larger) than 1/2, you could destroy (create) angular momentum without torque. Likewise for energy.
– JEB
Mar 9 '19 at 5:44
• @JEB Sorry, I read the title and wrote the answer, it's edited then. But just to see angular monument was not destroyed, it required addition condition to maintain the motion, the Hamiltonian/Lagrangian was changed.
– J C
Mar 9 '19 at 5:55

By definition moment of inertia of an object is calculated by cutting the object into increasingly smaller objects, or partitions, and multiply their mass by the square of their distance from the axis of rotation.

$$I= \Sigma \ m_i r^2_i = m_1r^2_1 + m_2r^2_2 +m_3r^2_3+ ...$$

Therefore no matter how many parts and in what random sizes we cut an object, as long as we do not disturb the original geometry its moment of inertia is not going to change.

For example If we cut a birthday cake into different random slices but don't move them apart, its moment of inertia will not change.

If we even move the parts but put them in a way that the new shape's radius of gyration , $$r_{gyration} = \sqrt{\frac{I}{m}} \quad ,$$ is the same as the original one the moment of inertia will not change.