May 8th, 2023
We once again start by establishing the same relationship between angle, angular velocity, and angular acceleration as we did for their linear motion counterparts. Then, using the fact that arc length is the product of angle in radians and radius, we can find the magnitude of the linear velocity at a point, with the direction being perpendicular to the radius. This can be proven by differentiating the displacement vector, as the velocity is just the derivative in respect to time of (rcosΩ, rsinΩ). Since we converted the angular velocity, we can now treat it as a linear velocity which can be used to find the total velocity of an object that is both rotating and translating. Like momentum in linear motion, angular momentum also sort of represents the amount of motion stored, but instead is calculated through the dot product of the perpendicular radius and momentum. In a sense, this calculation is determining how much of the linear momentum is in the direction of rotation, which actually allows us to derive the formula for torque given that force is actually the change in momentum over time.
Now the final piece of setup involves finding moment of inertia, a measure for how difficult it is to rotate an object, and acts in a similar manner to centre of mass as it streamlines the calculations. First, we can find the total angular momentum the same way as linear momentum, the summation of the angular momentum at every point. If we now take out angular velocity, we are left with the moment of inertia, the sum of the mass multiplied by the radius square of every point.
You now finally have all the background knowledge to fully simulate a basic 2d-physics simulation. First, you would calculate where the center of mass is, and the moment of inertia at that point. Then, you would find the sum of all the forces and divide by the total mass to find the linear acceleration. For each force, you would then calculate the torque through the dot product, then divide the total torque by the moment of inertia to find angular acceleration. With all that, you can then use integration on the two accelerations to find the actual movement of the body.