Stay home Friday
08-12-2016, 08:59 AM
#5
Moist IT Outsourcing Services
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08-12-2016, 12:48 PM
#14
Call me Pebbles
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Join Date: Nov 2011
Location: I do all my own physics.
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The “shape change” you see is a change in distribution of mass with respect to the axis of rotation. To dive (pun completely and unabashedly intended) into rotational dynamics, it helps to, first, define or introduce the rotational analogs to linear dynamics.
For linear or translational dynamics (motion in one direction/axis), there are two fundamental quantities: position (the object location in space) and mass (the sum total of the matter an object is composed of). From these two quantities, we can derive (again, pun fully intended) all other quantities and develop a set of equations to fully (to first principles) characterize an object’s motion. For simplicity, we will do this for a point-like mass or particle.
The rate at which the particle changes position, x, per unit time, t, is defined as speed, s, (more correctly, this is velocity, a vector, for 3D-space, but for simplicity we’ll look at 1D-space in which the velocity vector reduces to a scalar quantity, speed). The rate at which the particle speed changes per unit time is defined as acceleration, a. Thanks to good old Isaac, we know that the acceleration of object is directly proportional to the amount force, F, imparted on the object (the harder you hit it, the faster it moves), and inversely proportional to the mass of the object (the heavier it is, the more difficult it is to move). This is symbolically notated as a = F/m, or, more familiarly, F = ma.
Jumping over to rotational dynamics (motion about an axis), the quantities here are familiar, but slightly different (same circus, different clowns). Imagine the hand of clock face rotating about the center point. The direction the hand is pointing, or the location of the hand tip (just the tip) can be described by its angular position, θ (theta). This is the rotational analog to linear position. The rate at which the hand rotates about the center point per unit time is the angular velocity, ω (omega). This is the rotational analog to linear velocity. The rate at which the angular velocity changes (either speeds up or slows down) per unit time is the angular acceleration, α (alpha). This is the rotational analog to linear acceleration. Newton’s Second Law of (linear) Motion can be directly related to rotational motion by replacing the corresponding quantities for linear motion with their rotational analogs. The force analog acting on the rotating object is torque (yes, same torque as discussed when referring to engine output—the force of the outward accelerating combusting air/fuel mixture on the piston which transfers the force to the rotating crankshaft), τ (tau), and the mass analog of the rotating object is the moment of inertia, I. Rotational acceleration, α, is directly proportional to torque, τ, and inversely proportional to the moment of inertia, I—α = τ/I or τ = Iα.
Moment of inertia, or rotational inertia, must be specified with respect to an axis of rotation for an object or particle. If we take our point-like particle of mass m, and rotate it about an axis at a distance r from the axis of rotation (think Earth orbiting the Sun), the moment of inertia is defined as the product of the mass, m, and the square of the distance r: I = mr^2. If we take multiple point-like particles of varying masses and distances from the axis of rotation (think Earth, Mercury, Venus and Mars now orbiting the Sun), the total moment of inertia is equal to the sum of the individual rotational inertias: I(total) = I(Mercury) + I(Venus) + I(Earth) + I(Mars) = m(Mercury)r(Mercury)^2 + m(Venus)r(Venus)^2 + m(Earth)r(Earth)^2 + m(Mars)r(Mars)^2. This is where it starts to get messy, so try to keep up. If we were take an object with a continuous mass distribution, say an Olympic diver, we can quantify its rotational inertia by summing all of the rotational inertias for each small infinitesimal point mass throughout their body.
Here’s where the real fun begins. Remember back when we said that rotational inertia is analogous to mass and that acceleration is inversely proportional to mass (the heavier something is, the harder it is to move). If we take that same concept and apply it to rotational dynamics we can say that the greater an object’s rotational inertia, the more difficult it is to rotate. When the diver is at the beginning of the dive and they are tucked into a pike position, all the individual point masses in their body are close to the axis of rotation—remember, rotational inertia is directly proportional to the point mass distance from the axis of rotation so the closer the mass is to the axis of rotation, the lower the rotational inertia, the easier it is for the object to rotate. As the diver nears the water, they extend out their arms and legs effectively increasing the distance of their mass distribution from the axis of rotation, increasing their rotational inertia and slowing their rotational acceleration.
Here’s a fun experiment to illustrate rotational inertia for those of you with spinning office chairs. Get yourself spinning in your chair and try to keep your arms and legs close in. Once you get going, extend out your arms and legs, and see how your rotational velocity slows down. Now, bring your arms and legs back in, and your rotational velocity should speed up.
