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Competitive ice skaters commonly perform single, double, and triple axel jumps in which they rotate $1 \frac{1}{2}, 2 \frac{1}{2},$ and 3$\frac{1}{2}$ revolutions, respectively, about a vertical axis while airborne. For all these jumps, a typical skater remains airborne for For all these jumps, a typical skater remains airborne for about 0.70 s Suppose a skater leaves the ground in an "open" position (e.g. arms outstretched) with moment of inertia $I_{0}$ and rotational frequency $f_{0}=1.2 \mathrm{rev} / \mathrm{s},$ maintaining this position for 0.10 $\mathrm{s}$ . The skater then assumes a"closed" position (arms brought closer) with moment of inertia $I,$ acquiring a rotational frequency $f,$ which is maintained for 0.50 s. Finally, the skater immediately returns to the "open" position for 0.10 s until landing (see Fig. $49 ) .$ (a) Why is angular momentum conserved during the skater'sjump? Neglect air resistance. (b) Determine the minimum rotational frequency $f$ during the flight's middle section for the skater to successffully complete a single and a triple axel. (c) Show that, according to this model, a skater must he able to reduce his or her moment of inertia in midflight by a factor of about 2 and 5 in order to complete a single and triple axel, respectively.

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Physics 101 Mechanics

Chapter 11

Angular Momentum; General Rotation

Moment, Impulse, and Collisions

Rotation of Rigid Bodies

Dynamics of Rotational Motion

Equilibrium and Elasticity

Rutgers, The State University of New Jersey

Simon Fraser University

Hope College

McMaster University

Lectures

02:21

In physics, rotational dynamics is the study of the kinematics and kinetics of rotational motion, the motion of rigid bodies, and the about axes of the body. It can be divided into the study of torque and the study of angular velocity.

04:12

In physics, potential energy is the energy possessed by a body by virtue of its position relative to others, stresses within itself, electric charge, and other factors. The unit for energy in the International System of Units is the joule (J). One joule can be defined as the work required to produce one newton of force, or one newton times one metre. Potential energy is the energy of an object. It is the energy by virtue of an object's position relative to other objects. Potential energy is associated with restoring forces such as a spring or the force of gravity. The action of stretching the spring or lifting the mass is performed by a force which works against the force field of the potential. The potential energy of an object is the energy it possesses due to its position relative to other objects. It is said to be stored in the field. For example, a book lying on a table has a large amount of potential energy (it is said to be at a high potential energy) relative to the ground, which has a much lower potential energy. The book will gain potential energy if it is lifted off the table and held above the ground. The same book has less potential energy when on the ground than it did while on the table. If the book is dropped from a height, it gains kinetic energy, but loses a larger amount of potential energy, as it is now at a lower potential energy than before it was dropped.

01:39

A figure skater begins a s…

03:24

$\cdot$ The spinning figur…

05:00

I The spinning figure skat…

11:43

The Spinning Figure Skater…

04:08

The outstretched hands and…

11:58

Conservation of Angular Mo…

01:51

$\|$ Ice skaters often end…

13:25

In Fig. 11-47, two skaters…

during the jump. The Onley force on the skater is airborne, the only force on the skaters gravity because she is airborne. So when she is airborne, she's essentially a projectile in free fall. And so there's no there's no there's no there's not gonna be any force on the skater except for gravity. And this is acting through the center of Mass. So essentially, we can say that, um, there is no torque about the center of mass, and so we can say that angular mo mentum is conserved during the jump for Part B. We know that for a single axle, the skater must have one and 1/2 total revolutions. We know that the number of evolution's during each phase of the motion as the rotational frequency times the elapsed time. And so we can say that we also note that the rate of rotation is the same for both occurrences of the open position, so creating the relationship we have 1.2 revolutions per second multiplied by a point 10 seconds plus the frequency multiplied by 0.50 plus again, 1.2 revolutions per second multiplied by again point 10 seconds and we know that this must equal 1.5 revolutions. And so we can say that the frequency must equal 1.5 revolutions minus two times 1.2 revolutions, 0.10 seconds and then divided by 0.50 seconds. And so this is equaling approximately 2.5 revolutions per second. This would be, um, the frequency essentially for a single axle. And so the calculation would be the exact same for a triple axel. So continuing part B, we have 1.2 revolutions per second, multiplied by 0.10 seconds plus frequency triple axel times 0.50 seconds plus 1.2 revolutions per second multiplied by again 0.1 point +10 seconds. And this must now equal 3.5 revolutions because this is a triple axel. And so we find that the frequency would be equaling 3.5 evolutions, minus two times 1.2 revolutions for a second multiplied by 0.10 seconds. And then this would be divided. My 0.50 seconds. This is giving us approximately 6.5 revolutions her second. This would be our final, our second final answer for part B and then for part C, we're simply going thio upside the angular momentum, the conservation of angular ruin some to relate the moments of inertia. And so, for part C, we have moment of inertia, of the single open equals moment of inertia of the single closed. And this would give us the moment of inertia of the single opened, multiplied by the anger angular velocity of the single open equals moment of inertia. Single closed, angular velocity single closed. And so we find that the moment of inertia of the single closed, divided by the moment of inertia. For these single open, this would be Omega sub s O divided by Omega's of S C. This is equaling F sub s O divided by F. S. C. And this is equaling 1.2 revolutions for a second, divided by 1.52 revolutions per second will around at the very end. This is equal in point for seven six and we can essentially round to approximately 1/2. So this would be the ratio of the moment of inertia of the single close versus the single open. This is essentially me, meaning that the single axle moment of inertia must be reduced by a factor of two. Um, in order to do the triple axel calculation, it's the exact same thing. So you could say for part C moment of inertia for the triple axel closed position divided by the triple axel open position. This would be equal to F sub s. Sorry, have some tea. Oh, divided by f so tc and this is equaling 1.2 revolutions per second, divided by 6.52 revolutions per second. This is giving us 0.184 or we can say approximately the 1/5. So essentially, the triple axel moment of inertia must be reduced by a factor of about five in order to make the triple axel. So this would be your second final answer for part C. And this would be, ah, your first answer for part C. That is the end of the solution. Thank you for watching

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