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Hello

My student is exploring the mathematical equations in the physics side of movement (especially dance

The essential question of her project: What mathematical equations can I use to find positioning of the body that will improve ease / proficiency of movement?

She was hoping to look into this question through three types of movements:
1. Figure skating - if you are doing a spin on ice and you leave your arms spread out it creates resistance, if you pull the in towards the body you create less resistance so you spin more easily. How do I express this with mathematical equation in physics?

The other two areas she might want to explore other equations will relate to martial arts and dance. But if you can help her with the ice skating question that will be wonderful.

Her final exhibition (we use these instead of tests) is on 3/7 so she needs to hurry up with her plans.

Many thanks,
Zohara

Answer
Hello Zohara,

I can explain the physics of an ice skater's spinning. But regarding martial arts and dance, I would need more detail about the movements you would like me help you understand. If you could be more specific in a follow-up question regarding motions of martial arts and dance I would try to help you understand that also.

The principle of conservation of momentum is the reason a skater's spin speeds up when she pulls in her arms. First I need to discourage the idea that the increase in spin speed when pulling in the arms has to do with decreasing air resistance. The affect of air resistance is to cause the skater to slow down. Having the arms pulled in will decrease the air resistance and therefore decreases the rate that the skater slows down because of air resistance. But it does not cause her spin to speed up. The increase in speed when the arms are pulled in is due to the principle of conservation of momentum.

Momentum is seen in straight line motion and in rotation. Let me talk about straight line motion first. Isaac Newton's 1st Law says (among other things) that an object in motion remains in motion, in a straight line, unless acted on by an outside force.

If you are against a wall and you give a hard push to a fellow skater, that skater can allow the momentum that your push gave her to carry her across the ice. It is friction from the skates on the ice and air resistance that eventually slows her to a stop. Even if she could make herself more streamlined, the skater would need an additional push, an outside force, in the forward direction to increase her straight line speed.

In straight line motion, the formula for momentum is
p = m*v
Both the amount of mass and the value of the velocity contribute to a body's momentum. So an object with large mass moving slowly can have the same momentum as a lesser mass moving faster. The momentum that applies to a spin is called angular momentum.

There is a version of Newton's 1st Law that applies to a spinning body. Let me say it this way: an object that is spinning continues to spin unless acted on by an outside torque. (A torque is like a force except it is delivered in a way to cause or to stop a spin.) I mentioned conservation of momentum above without much detail. The principle of conservation of momentum applies to the type of momentum of an object with straight line motion. That principle is involved in the study of collisions. The principle of conservation of momentum also applies to the type of momentum of a spinning skater. That type of momentum, called angular momentum, is more complicated than straight line momentum. How the total mass and how the mass is distributed is part of the study.

The formula is
L = I*w
where I is rotational inertia and w is angular velocity. It is the rotational inertia term that considers the amount of mass and how it is distributed. Mass that is close to the axis of rotation has less of a contribution to the value of a body's rotational inertia, or I, than mass that is extended away from the axis of rotation. In fact the contribution to I of a particular increment of mass is proportional to the square of its distance of extension. So if you start with your hand pulled in and then extend your hand so the distance from the axis of rotation is tripled, the hand's contribution to your body's total value of I increases by a factor of 9.

Once the skater has started spinning, angular momentum is conserved. The principle of conservation of angular momentum says that the value given by the formula L = I*w must remain constant even when the skater suddenly pulls in her arms causing the value of I to suddenly decrease. The value of L must remain constant, so the value of w must suddenly increase. When the skater pulls in her arms, the value of I decreases so the value of w (the rotational speed) increases to allow the value of L (the angular momentum) to remain constant.

I have found some web sites that give different ways of explaining this. It may help to visit these sites:
http://www.real-world-physics-problems.com/physics-of-figure-skating.html
http://www.bsharp.org/physics/spins
http://iceskatingresources.org/PhysicsOfSkating.html
http://www.livescience.com/6120-physics-figure-skating.html

I hope this helps,
Steve

Physics

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Steve Johnson

Expertise

I would be delighted to help with questions up through the first year of college Physics. Particularly Electricity, Electronics and Newtonian Mechanics (motion, acceleration etc.). I decline questions on relativity and Atomic Physics. I also could discuss the Space Shuttle and space flight in general.

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I have a BS in Physics and an MS in Electrical Engineering. I am retired now. My professional career was in Electrical Engineering with considerable time spent working with accelerometers, gyroscopes and flight dynamics (Physics related topics) while working on the Space Shuttle. I gave formal classroom lessons to technical co-workers periodically over a several year period.

Education/Credentials
BS Physics, North Dakota State University
MS Electrical Engineering, North Dakota State University

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