# Reasons for using Angular momentum, Torque and Moment of inertia to describe rotational motion

In order to describe rotational motion, we usually ditch the familiar concepts of force, linear momentum and mass, and use instead their moments to describe the motion. Is it just because it makes our calculations easier, or is there a deeper reasoning behind it.

In case of translational motion, where we work with force, mass and momentum, the shape of the body doesn't matter. However, in case of rotation, it seems that the shape of the body does matter, and treating them as point masses would give us incorrect results.

For example, consider a physical pendulum or a compound pendulum. Instead of using force or mass, we try to analyze it using torque/moment of inertia. Is it because the latter considers the distribution of masses, while the former ignores it ? Or is there a deeper theoretical reason for using the moments.

So, if we used force/mass to analyze a compound pendulum, we would get a wrong answer for it's time period, if we compare that to experimental answers. However, the answer we get by analyzing torque would be closer to the experimental ones. Is that the reason we use Torque to analyze rotation - because it considers the distribution of masses, and thus agrees more with experimental results, or is there some theoretical reasoning behind it, and we can just as well describe any compound pendulum, purely using forces ?

The concept of moment of inertia arises naturally as soon as you consider the rotation of a rigid body about a fixed axis - or even just a set of particles moving with the same angular velocity about a fixed axis.

For example, just consider the kinetic energy of a rigid uniform rod rotating with angular speed $$\omega$$ about a fixed axis at one end. If you isolate one small element of mass $$m$$ which is situated at a distance $$r$$ from the axis, its kinetic energy is $$\frac12mv^2=\frac12m(r\omega)^2=\frac12(mr^2)\omega^2$$

Every such particle has the same angular velocity but are situated at different distances from the axis. To get the total kinetic energy you need to sum this expression over the whole body, i.e. $$KE=\sum\frac12(mr^2)\omega^2=\frac12\left(\sum mr^2\right)\omega^2$$

Therefore we need to be able to calculate the expression $$\sum mr^2$$ for the whole body in relation to this axis i.e. the moment of inertia.

The resulting expression would not be the same as considering the whole body as a particle of the same total mass concentrated at the centroid, and experimental results would reveal the inconsistency.

• This is exactly the answer I was writing. Nicely done.
– rob
Nov 1, 2021 at 11:29

Let's assume that you are dealing with a rigid body that can rotate about a given axis. That means that if you displace the body all particles in it have the same angular displacement about that axis. You are immediately urged towards considering a rotational approach.

I always called linear motion and rotational motion "sisters". For example, you have the linear Force, and its rotational sister torque.

You actually need both to analyze the motion of an object. Here's an example. It's just that on classes, you are being taught one or the other, so the problems you are given only include only linear or only rotational motion.

Extreme answer: I mean, if you are a truly mad scientist, you could use linear forces on your body's molecules and also consider cohesion forces. There is no reason to use this, though. The rotational equations are a nice and working encapsulation of all that stuff.