Re: nunchuck physics

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Two rigid rods, connected by a short rope? I don't see the whip model, and I don't see a nunchuck 'waving'...
However, that does lead me to ask: I have seen reports that a whip's tip will move at near speed of sound, when it 'cracks'. It's the same mystery as my nunchuck question - how does that speed arise? You can't explain it in terms of rotation and arm geometry...
Sam

"Sam the Bam" wrote You're thinking too large. A whip is just a multi-segment nunchuck. If you had a million tiny rigid segments tied together by a million tiny ropes, it would look and act just like a whip.
It's just the kinetic energy of the whole whip motion being concenctrated in to the whip tip. The energy comes from the user's arm...the rest of it is just a fancy energy/motion converter that takes slow, large-displacement motion of one end of the whip and concentrates it to very fast, small-displacement motion at the end. Sort of a solid-state gearbox, if that makes any sense.
Of course not. A whip (and a nunchuck) isn't a rigid body.
Tom.

actually, you can - it's just that energy methods are easier.
Think of you standing on a spinning schoolyard merry-go-round. When you move your weight towards the center of the moving merry-go-round, it speeds up due to conservation of angular momentum,
i.e., because the radius between your mass and the center of the merry-go-round shortens, the rotational velocity must increase in order to have the same momentum as when you were farther out.
So when you cause the handle end of a bull whip to turn in an arc, you have "spun the merry-go-round" with the whip mass and arc diameter -- and as the arc travels down the whip, the arc gets smaller because the whip is smaller in diameter and thus its mass in the arcing portion gets smaller, that smaller mass allowing the arc to be smaller, making the "end" nof the arc section move faster due to conservation of momentum - and faster and faster as the arc moves along the thinner and thinner whip. Until it reaches the end and suddenly reverses the tip as the wave reflects.
A horse-whip (flexible length on a rigid stick) sends its energy into the flexible section, amplifying the user's arm motion by effectively extending the user's arm - a sudden reversal of the stick causes a wave, the wave reverses at the tip, and conservation of momentum in the manner above causes the tip to move very fast and crack.
A nun-chuck is a common older farming tool across the world. It is used to break the grain hulls off the kernel. AKA known as a flail.
The momentum in the stick-link-stick is developed as if it were one stick, and is stored in the three parts during the initial stroke. Stopping the handle forces the developed rotational momentum into the other two parts - the link facitiltates transfer of momentum of the long three-piece stick into the end stick and stores little itself, i.e., the original rotational momentum is transferred into one-third the original size end stick.
The short end stick is then travelling in a smaller arc, conserving the momentum of what was just effectively a long stick in a smaller mass AND a smaller arc.
And on top of the conservation-of-angular-momentum increased velocity of the end stick over that of a long single-piece stick, energy is then highest in the tip of the end stick, since it moves the fastest in an arc ( from ke=1/2 m V^2, )
fwiw

I was thinking about this some more, and wonder: would a nunchuk perform differently, if it had a hinge, in lieu of a rope?
Sam

But you don't. You have 2 rods, with one rope. The limit argument doesn't apply.
That seems clear, but...
It doesn't. The "concentration of energy" pseudo-explanation simply begs the question.
Ditto the gearbox model - gear motion is simply explained, refering to sprocket radii as moment arms; a wheel size ratio 2:1 produces linear velocity 2:1.
Sam

I don't follow this. The whip's mass is moving toward the handle, like your merry-go-round analogy? What does it mean, "whip is smaller in diameter"?
Is it crucial that the whip tapers to the end? Because that's not the case in a nunchuk...
A pity we can't draw diagrams in this medum...
I think I get this. Angular momentum is reduced in the handle, which by conservation, must appear in the load stick. I have trouble picturing how this energy transfer occurs, though... it's not a wire conducting electricity, or heat diffusion...
And logically, per this argument, the handle must be decelerated to accelerate the load - which means no velocity gain occurs while the user twirls the handle at constant speed? That contradicts perception, if not reality...
Sam

no

momentum is built up - it remains in the system and is redistributed by slowing the handle
which means twirl the handle at constant speed and watch the end - there is no velocity gain.

Tp make something move quickly, you need to provide kinetic energy. Work done is force x distance, or torque x angle rotated through.
Take a brick, and see how far you can throw it. The range is approximately proportional to v^2, so proportional to energy/mass. Take a baseball, and see how far you can throw it. Compare the range vs energy/mass. Take a marble, and do likewise. Finally, try a ball-bearing.
There's a limit to how fast you can move your hand. For throwing, the limit applies to how fast you can move your hand forwards during the throw. For twirling a stick, it depends on how fast you can turn you hand in a circle. If, instead, you twirl a weight a string, you no longer need to
It isn't a schoolboy physics problem. It isn't basic physics - it's biomechanics.
Will the tip of the stick increase in speed? Perhaps, perhaps not - it depends on where the centre of mass is compared to your grip. Try it with a well-weighted stick.
With the stick (or better, something balanced like a sword, like a sword), the point about which the rotation takes place is your grip. With nunchucks, the rotation takes place about the point where the rope attached to the part you're holding onto.
Angular momentum will be conserved. The exact details of what happens depend on the balance of the nunchucks. Having the centre of mass closer to the pivot point gives a larger increase in speed (and as a result, makes the nunchucks harder to control, and isn't always desirable).
OK, let us do some maths for the simplest case: nunchuck moving forwards at speed v1, with no rotation at all. The held half is stopped abruptly, so the free half swings around. There will be a force applied on the free half at the pivot point which will slow the foward motion of the centre of mass of the free half, but as this force goes through the pivot point, there is no torque about this point. Therefore, the angular momentum about the pivot point will stay the same.
Let: L = length of free half L/2 = distance to centre of mass v1 = initial speed v2 = final speed of tip, which we are trying to find m = mass of free half I = moment of inertia of the free half, = mL^2/3 for rotation about one end w = angular speed, = v2/L
Initial angular momentum = linear momentum x L/2 = m v1 L / 2
Final angular momentum = I w = m L^2 v2 / 3L = m L v2 / 3
Since this is equal to the initial angular momentum,
m L v2 / 3 = m v1 L / 2 v2 = 1.5 v1
so the tip moves faster. (The centre of mass moves slower, since its final speed is v2/2 = 0.75 v1)
The example with the maths above is about as far as you can go with Newtonian physics. Beyond that - and for the swing-around-many-times problem, you're in biomechanics, which is rather harder than basic Newtonian physics.