The Mathematics of

1. The Mathematics of Ice Skating soniaburney tanyahou sybillam andreaolarig

2. Background Information • a double axel and a split jump create a • the moment of inertia while spinning changes as arms are brought towards the body • maximum potential energy is reached at the maximum vertical height of a jump • at the beginning and end of a jump, parabolic path maximum kinetic energy is reached

3. Kinematics assuming no air resistance or friction • s • skaters must be concerned with • skaters try to achieve maximum horizontal and vertical displacement during their jumps • unlike speed skaters and ski jumpers, ice skaters generally are not moving fast enough to have their jumps be affected by air resistance • the maximum height of jumps, rotational speed, and horizontal speed center of gravity always follows a parabolic shape Y X

4. Kinematics and Ice Skating • the take-off angle take-off velocity, height of take-off are the three factors which determine the figure skater's trajectory during a jump • (it is very important to separate the object's vertical take-off velocity from their horizontal take-off velocity) • once gravity has slowed the skater's upward vertical velocity to zero, gravity then accelerates the skater back to the earth • velocity is ZERO at the peak of the jump/vertical displacement is of maximum value take-off angle take-off velocity height of take-off velocity is ZERO

5. initial velocity final velocity constant acceleration Kinematics Equations position

6. Calculus of Kinematics

7. Application of Kinematics • projectile motion • Donald Duck is about to perform a double axel. He takes off at an angle of 45° and his vertical displacement is 1 meter and his horizontal displacement is 2.5 meters. If his flight time is 0.5 seconds, what is the initial velocity of his jump? [ignore rotation]

8. x component y component continuedpartI

9. continuedpartII • so, using kinematics we can find the initial and final velocity in the x direction of the figure skater given specific quantities

10. Rotation [Moment of Inertia] • assuming no air resistance or friction • rotational inertia is the resistance to change in rotation • scratch spins: the axis of rotation is the body of the skater. • the rotational inertia of a given mass is given by • inertia also equals • the rotational inertia increases as the square of the distance to the axis • for example, if distance is doubled, then inertia is quadrupled. Even a small change in position can affect the rotational inertia rotational inertia is the resistance to change in rotation the rotational inertia increases as the square of the distance to the axis

11. The Moment of Inertia • the moment of inertia of a spin is increased as arms are brought out since the radius of the spin increases • the radius and the moment of inertia are directly proportional the radius and the moment of inertia are directly proportional radius of arms is large and so inertia is also large since radius is small, inertia is small and so the skater is less likely to resist change in rotation

12. Application of Inertia • moment of inertia for a skater • the mass of our skater will be 50 kg. Her torso and standing leg are about 40 kg. The radius of the skater will be 0.1meter. Her torso and one leg will be represented by a cylinder so the moment of inertia will be • this is the moment of inertia around her torso • and standing leg

13. continued • now, we have to find the moment of inertia for her arms and other leg as they move closer to the body • in this case • this is the moment of inertia of her arm and leg • this diagram represents the basic inertia of the skater (central axis) and the circle she creates • to find the total moment of inertia, we can add the inertia of the torso/leg with the inertia of the arms/leg

14. any change in rotation is due to a force Rotation [Angular Momentum] • angular momentum is an object’s resistance to change in rotation • angular momentum is conserved if there is no force present • the force needed to create angular momentum is produced when the ice skater pushes against the ice • angular momentum is defined by an object’s resistance to change in rotation conserved if there is no force present

15. Calculus of Angular Momentum • since angular momentum is conserved with no force, if changes then must change so stays constant at various time intervals • as the moment of inertia decreases, angular velocity increases • rotational kinematics for spins as long as is constant • angular velocity • angular acceleration • angular momentum

16. Application of Angular Momentum • we can use rotational kinematics to find angular momentum • position • since we can take the derivative of position at a given time to find angular momentum. • for example, if initial velocity of a spinning skater is 20 rad/sec with a constant acceleration 10 rad/sec², at time = 0.1 sec, what is the angular momentum of the spinning skater? (using inertia found from pervious application)

17. Rotation [Calculus of Torque] • torque is the type of force that makes something rotate • there is no net torque when angular momentum is conserved • the equation for torque is defined by • since during a scratch spin, angular momentum changes, so there is a net torque acting on the skater • also or

18. Application of Torque • so, given a specific equation at a specific time and inertia, one could find the net torque • for example, at t = 0.2 sec given • And, we can find power of the skater at t =0.2 sec

19. Energy • to better understand how potential and kinetic energy works, rotation and torque will be negligent • the skill that also best fits this type of criteria would be the split jump • something to always keep in mind is that energy is always conserved and neither created nor destroyed

20. Starting with Potential • as with any jump or skill, the skater needs to build up or increase the amount of potential energy they have before executing a move • since by definition potential energy is independent of motion • this means that the potential energy of a skater is the amount of energy stored in muscle power • after a skater jumps, their muscle power potential energy is converted to kinetic energy, which is then converted to gravitational potential energy (at the very top of their flight), and then converted back to kinetic once they land by definition potential energy is independent of motion

21. mechanical energy potential energy kinetic energy Kinetic Energy • if potential energy is the amount of energy when an object is not in motion, then kinetic energy is the energy an object has by virtue of its motion • the sum of kinetic and potential energy is mechanical energy • assuming there are no nonconservative forces (such as friction), then mechanical energy is conserved since energy is always conserved – Law of Conservation of Total Energy • this basically means that the initial mechanical energy of the skater is equal to the final mechanical energy kinetic energy is the energy an object has by virtue of its motion Law of Conservation of Total Energy

22. Further into Energy • How does potential and kinetic energy all relate to the skater? • First of all, potential is • This would be applied as the mass of the skater multiplied by gravity (a constant) multiplied by the skater’s height

23. continued • next, kinetic energy is • where is this derived from? • since the skater’s acceleration is and will have traveled a distance of then the final speed, can be proven with kinematic equations • one might notice though, that • however, work done by the skater has transferred energy to it, which comes in as kinetic energy

24. Application of Energy • a skater is coming out of her split jump at a 20° angle, and lands skating on the ice a distance of 10m. If the coefficient of kinetic friction of the skates and ice is 0.2, calculate the skater’s speed at the end • the strength of the friction force on the skater is • so work done by friction is • the vertical height in the end would be calculate the skater’s speed at the end

25. continued • hence, with the Law of Conservation of Total Energy, this gives us

26. Hokin, Sam. “Figure Skater Spins.” Online. Available: http://www.bsharp.org/physics/stuff/skater.html, 14 May 2006. King, Deborah. “The Science of Jumping and Rotating.” Online. Available: http://btc.montana.edu/olympics/physbio/biomechanics/bio-intro.html. 14 May 2006. Knierman, Karen and Rigby, Jane. “The Physics Ice Skating.” Online. Available: http://satchmo.as.arizona.edu/~jrigby/skating/main.html, 14 May 2006. Nave, C.R. “HyperPhysics.” Online. Available: http://hyperphysics.phy-astr.gsu.edu/HBASE/hframe.html, 14 May 2006. Works Cited