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The Mathematics of soniaburney tanyahou sybillam andreaolarig Ice Skating

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

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

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Kinematics and Ice Skating othe take-off angle take-off velocity, height of take- off are the three factors which determine the figure skater's trajectory during a jump o(it is very important to separate the object's vertical take-off velocity from their horizontal take-off velocity) oonce gravity has slowed the skater's upward vertical velocity to zero, gravity then accelerates the skater back to the earth ovelocity 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

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Kinematics Equations position initial velocity final velocity constant acceleration

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Calculus of Kinematics

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Application of Kinematics oprojectile motion oDonald 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]

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continuedpartI x componenty component

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

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Rotation [Moment of Inertia] oassuming no air resistance or friction orotational inertia is the resistance to change in rotation oscratch spins: the axis of rotation is the body of the skater. othe rotational inertia of a given mass is given by oinertia also equals othe rotational inertia increases as the square of the distance to the axis ofor 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

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The Moment of Inertia othe moment of inertia of a spin is increased as arms are brought out since the radius of the spin increases othe 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 the radius and the moment of inertia are directly proportional

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Application of Inertia omoment of inertia for a skater o 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 o this is the moment of inertia around her torso and standing leg

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

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

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

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Application of Angular Momentum owe can use rotational kinematics to find angular momentum oposition osince we can take the derivative of position at a given time to find angular momentum. ofor 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)

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Rotation [Calculus of Torque] otorque is the type of force that makes something rotate othere is no net torque when angular momentum is conserved othe equation for torque is defined by osince during a scratch spin, angular momentum changes, so there is a net torque acting on the skater oalso or

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Application of Torque oso, given a specific equation at a specific time and inertia, one could find the net torque ofor example, at t = 0.2 sec given o And, we can find power of the skater at t =0.2 sec

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

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Starting with Potential oas with any jump or skill, the skater needs to build up or increase the amount of potential energy they have before executing a move osince by definition potential energy is independent of motion othis means that the potential energy of a skater is the amount of energy stored in muscle power oafter 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

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Kinetic Energy oif 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 othe sum of kinetic and potential energy is mechanical energy oassuming there are no nonconservative forces (such as friction), then mechanical energy is conserved since energy is always conserved – Law of Conservation of Total Energy othis 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 mechanical energy potential energy kinetic energy

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Further into Energy oHow does potential and kinetic energy all relate to the skater? oFirst of all, potential is oThis would be applied as the mass of the skater multiplied by gravity (a constant) multiplied by the skater’s height

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onext, kinetic energy is owhere is this derived from? osince the skater’s acceleration is and will have traveled a distance of then the final speed, can be proven with kinematic equations oone might notice though, that ohowever, work done by the skater has transferred energy to it, which comes in as kinetic energy continued

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Application of Energy oa 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 othe strength of the friction force on the skater is oso work done by friction is othe vertical height in the end would be calculate the skater’s speed at the end

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ohence, with the Law of Conservation of Total Energy, this gives us continued

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

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