By Eugene Toth
Bodies that have rotation have rotational inertia. They will continue to rotate around axis unless stopped by an outside force. For example, the Earth rotates around the Sun. Because of rotational inertia, astronauts are exceedingly careful to avoid getting into a spin.
Just as there are linear and rotational velocities, there are also linear and rotational accelerations. As we use the Roman letter α for rotational velocity, we also use it for rotational acceleration. Just as linear velocity is equal to angular velocity times the distance from the center of rotation. Therefore linear acceleration is equal to angular acceleration.
By the second law of motion, we know that force = mass x linear acceleration. Let a represent angular velocity. Let r represent rotation. “τ” is the greek letter “tau.” In our case, we will use it to symbolize “torque.” This formula gives us:
F = M r a
Force = Mass x Rotation x angular velocity
Force = mass x radius x alpha
Therefore we can say that the force of torque is mass x radius x alpha. We can therefore make the equation
Force = m r α
Then if Torque = force x radius, then
τ = m (r x r) α
τ = m r2 α
τ = 9 x 4 x 12
τ = 432
τ/α = m x r2
432/12 = 9 x 4
36 = 9 x 4
36 = 36
rotational acceleration = mass x radius 2
In linear motion, all components making up the mass of a an object are equally involved in an acceleration no matter what the shape of the object. For example, you can have the mass of 1000 kilograms, but made up of 1000 blocks of 1 kg each. You would need the same force to overcome its inertia, whether to create motion or to stop it. But for rotation, this statement does not apply
Rotational motion increases from center to periphery. It is not mass alone that counts, but mass x a square of the distance from the center.
Imagine a wheel composed of 1000 kg units evenly distributed. It is made out of 1000 sub units. Some are close and some are far. Those close to the axis have a small r and a small m r2. the larger units have a large radius and a large m r2.
The body as a whole has some average m r2. This is called the “Moment of Inertia” This can be symbolized by I. The torque required to stop a rotating wheel depends on I and not on mass.
Consider two spheres of equal mass but different distribution of density. Even if these two have the same mass, the one with the dense core and a light mantle will have a smaller moment of inertia I than the one with no core.
Imagine we have 2 spheres or wheels both composed of part metal and part plastic and of the same mass. One has the metal inside and the plastic outside. The other with the metal outside and the plastic inside and the plastic inside. The value I of each object would be quite different.
Consider a basketball. The inside is full of air with very low density. The outside has a very high density and a thin layer. It has a high I in relation to its low mass which gives it a high angular momentum.
Flywheels and gyroscopes are so designed and constructed to have a high rim density and a low density inside.
Just as there is a law of conservation of linear momentum, so there is a law of conservation of angulation momentum. The total angular momentum of an isolated system of bodies remains constant.
Ordinary momentum is mass x velocity. Can this formula apply to angular momentum?
No, angular velocity is not the same as ordinary momentum because inertia depends on mass x radius, not mass alone.
The symbol for angular velocity is Ω. I Ω represents angular momentum. Now we come to a strong phenomena.
Imagine that a man is standing on a spinning turn table, that is, turning on a bearing producing almost zero resistance.
Imagine that a man is holding two heavy weights by his side on a turning table. Now imagine him extend his arms out from the side of the body, turntable will slow. If he returns his arms to his side the turntable will accelerate. What is happening?
Answer: Consider the man on the turntable with the weights in his hands an isolated system. The force of acceleration is assumed to be present in the turntable and transmittable to the man and his weights. We are not adding or taking any energy from the system. When the man extends his arms, the value mr2 increases. If the same speed were to be maintained additional energy would have to be put into the system. As this is not done, the system must slow down.