Thursday, June 28, 2018

Rotational kinematics - possible and impossible spin-ups

In this first case as pictured below, an ice skater takes advantage of the conservation of angular momentum to increase rotational velocity by bringing their arms in. However, forces and torque need to be applied for this to work. The forces must also be balanced with respect to the axis of rotation, otherwise the axis would precess.

A naïve understanding of the forces and torques at play would lead one to believe that pairs skaters (or two co-rotating comet lobes) could come together and corotate quicker due to the conservation of angular momentum. That understanding would be wrong. The main issue is that in the single skater case, the force and torque is made from the outside of the centre of mass. When two co-rotating skaters pull in to each other, there is no way to exert the right forces on each other or the ensemble to torque to the higher speed required for continued co-rotation at a reduced moment of inertia.

This is the same with two co-rotating lobes of a comet or asteroid - neither gravity nor friction (at their points of contact) gives forces in the right direction to provide spin up torque.

Resultant endpoints vary depending on the reaction forces and friction, but none of them end up with the same co-rotating lobes at a faster speed and smaller net moment of inertia (or co-rotating couple at faster speed) 

Monday, June 11, 2018

Why couples figureskaters (and bi-lobe comets) don't do that spin up trick that individual iceskaters do!

It is well known that a figure skater takes advantage of the conservation of angular momentum to achieve incredible speeds of rotation by having their arms outstretched and bringing them in.

Ever wonder why they bring both arms simultaneously?
Or why couples skaters don't/can't pull each other in to achieve spectacular combined rotations?
or why bi-lobe comets don't collapse on the neck and reconfigure to a much faster combined rotation?

In short, the problem is the Coriolis force!
If an ice skater has just one arm outstretched and pulls it in at a rotating reference, the force required to counter the centrifugal force would topple the skater with the skates as the fulcrum and they would precess violently. The equal and opposite arm acting in unison balances the forces required to bring the arms in and all the work exerted on doing this gets converted to torque, spinning up the skilful skater.

With the couples skating this becomes a two body problem. As they pull each other in from outstretched arms, each person's individual moment of inertia is far more than the nominal amount of a single outstretched arm. Therefore, the radial pulling that they can exert on eachother  topples them both over quite effectively. The only way they can exert the torques required to hold them in a configuration is to wrap their arms around each other. Countering the centrifugal force and friction at their contact points is not enough to effect the principle, as it CANNOT prevent the two persons from spinning independently from each other, conserving angular momentum both individually and as a non rigid unit.

This principle, when applied to a bi-lobed comet like 67-P Churyumov-Geramisenko means that a partial collapse of the neck which allows mutual gravity to pull them closer together could not result in a faster rotation keeping the same configuration in the rotating reference frame.

 However, if the lobes rocked at the neck with a slight precession, the same configuration could be kept and the rotation restabilised with less precession if the neck lengthened and the friction caused a slower mutual rotation in the same configuration in the rotating reference frame.

This is exactly the same principle that couple skaters use to stabilise the rotation when doing death defying spins with the female skater's head just inches above the ice. A gradual lengthening of the arm under tension dampens any wobble in the spin allowing complete and graceful control of the spin.