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Angular Momentum
• Angular momentum is the momentum of rotation, and in this case refers to an object (the skater) with a fixed mass rotating on a fixed axis • Skaters must generate angular momentum to complete any spin or jump • In a spin, angular momentum is lost due to friction, and no matter how a skater may alter their body position, eventually they must stop • Angular momentum is the momentum of rotation, and in this case refers to an object (the skater) with a fixed mass rotating on a fixed axis • Skaters must generate angular momentum to complete any spin or jump • In a spin, angular momentum is lost due to friction, and no matter how a skater may alter their body position, eventually they must stop • To put this into perspective, the Top Fuel drag racing world record of 4.4s over 1/4 mile is 4.2 G's, but this acts upon the whole body, not just the arms!! • Angular momentum is conserved, which means that an increase in angular velocity (such as when skaters bring their arms and legs in during a jump) results in a decrease in the moment of inertia (and vice-versa) • In the above formula, angular momentum is L, inertia is I and angular velocity is lowercase omega.
Torque
Improvements
• Torque is a twisting force acted upon from a certain distance. •Skaters use torque to generate angular momentum in any spin or jump with their blades. • In either case, the skater applies a force on the ice, and the ice applies a force back on the skater. • It is this reaction force that is the torque used to generate angular momentum. • Greater force equals greater torque. • Similarly, greater torque equals greater angular momentum. There are 3 rules of torque as it applies to figure skating: 1) The force must be applied at a certain distance from the skater's centre of mass i.e., from the skate blades • Greater distance equals greater torque, so skaters with longer legs have a slight advantage 2) Also, the line of action must be outside the skater's axis of rotation, or the skater will fall • Think of it this way: if someone pushed you in the stomach on the ice, you would fall • However, if someone pushed your extended arm, with good balance you would be able to spin on the ice 3) Since torque is a twisting force, there must be some pivot point; in spins and jumps that do not use the toepick (like salchows, loops and axels), this is just behind the toepick, on the ball of the foot; and in toepick-assisted jumps (like toe-loops, flips and lutzes), this is the toepick. • Torque is represented by the greek letter tau, r is the length of the lever arm connecting the axis to the point of force application, and uppercase theta is the angle between the force vector and the lever arm vector. • In conclusion, skaters must make use of torque to generate angular momentum, otherwise they will not rotate.
• Inertia is an object's tendency to keep on doing what it is already doing: Newton's 1st law • Moment of inertia is basically a measure of the distribution of mass from the axis of rotation. • Inertia itself is not a definitive quantity, but rather a principle; however, moment of inertia is a quantity and can be derived from measured values. • An increase in inertia will result in an increase in angular momentum (angular velocity is held constant) • This is why skaters start any spin or jump with their arms and free leg extended; to keep inertia high • If a skater starts a spin or jump with their arms tucked in close, inertia is less and so is angular momentum (while angular velocity is held constant), which means that they won't be spinning very long and their jumps will undoubtedly be underrotated, especially the harder jumps • If a skater wanted to increase their angular velocity, which they do when they are performing almost any jump or spin, they would be better off starting with arms and free leg extended •This is due to the law of the conservation of angular momentum (angular momentum is always conserved in a frictionless environment), and when inertia goes down, angular velocity goes up • This principle was used when a group of North American figure skaters were trying to figure out how to land a triple axel to keep up with the Russians. • Through careful video analysis, they found out that the skater stays in the air for the same amount of time as the single or double axel. • To get the extra half-revolution, the Russian figure skaters were getting into their tight jumping position more quickly, thus making the best possible use of their angular momentum; by reducing their inertia and increasing their angular velocity. • In conclusion, high inertia can be exploited to help increase angular velocity when needed and pull off some of the most spectacular tricks of the sport: the combination spins and the double, triple and even quadruple jumps.
• Angular velocity is the number of revolutions an object makes per unit time - how fast it is spinning • A common unit for angular velocity is rpm, or rotations per minute • The SI unit for angular velocity is rad/s (radians per second), but because figure skaters rotate on a fixed axis, rev/s (revolutions per second) is used instead • Figure skaters must rotate very quickly to perform their tricks, especially the jumps • In fact, some of the world's best skaters reach rotation speeds of 7 rev/s during their jumps • This corresponds to 420 rpm, which is as fast as the idling speed of the engines of some cars!! • Angular velocity is typically represented by lowercase omega (as seen in the formula for angular momentum). • In conclusion, there are three kinds of velocity that are required for any well-executed jump: angular, vertical and horizontal. The latter two will be discussed in the next section.
Angular Velocity
The Physics of Figure Skating
1) WHAT IS PROJECTILE MOTION? • Projectile motion is the type of motion that any object follows through the air when there is no applied force (e.g. javelin toss, baseball throw, etc). • All figure skating jumps follow a path of projectile motion • In the case of figure skating jumps, the motion path is always a parabola. • This is because the only force acting upon the skater is gravity. • The take-off angle, the take-off velocity, and the height of take-off all affect the trajectory that the skater follows through the air, 2) VERTICAL AND HORIZONTAL VELOCITY • Since gravity affects only the vertical velocity, the skater is always at the top of the jump when the vertical velocity reaches zero, • However, vertical and horizontal velocities are not measured, but rather derived from the take-off velocity (also called resultant velocity) and the take-off angle. • Vector addition can be used to calculate these, and SOH-CAH-TOA (primary trigonometric ratios) are the easiest way to do that. • For example, let's say the take-off angle is 45 degrees and the take-off velocity is 10 m/s • The vertical velocity can be calculated as shown below, where Vto is take-off velocity, Vv is vertical velocity, Vh is horizontal velocity and (-) will be the angle (uppercase theta). A dot will be used to represent multiplication: Vv = Vto•sin(-) Vh = Vto•sin(-) •Therefore, by substituting the angle of 45 degrees and the take-off velocity of 10 m/s, we can conclude that the vertical velocity is about 7.1 m/s and that the horizontal velocity is also around 7.1 m/s. 3) VERTICAL DISPLACEMENT • Let's say we wanted to calculate the vertical displacement, and we were given the vertical and horizontal velocities of 7.1 m/s from the previous calculation. First we would want to calculate the time up (the time it takes the skater to get to the top of the jump) by using the formula below, where Vf is the final velocity, Vi is the initial velocity, a is the acceleration due to gravity (-9.8 m/s^2) and t is the time, which we are solving for: Vf = Vi + at • The final velocity is the velocity ot the top of the jump, which is always zero as discussed previously, the initial velocity is the vertical take-off velocity and, therefore, the time-up is 0.72 seconds. • Using this, we can calculate the vertical displacement using this formula, where D is the vertical displacement and all other variables are the same as above: D = Vi•t + 1/2 at^2, which can be used for any object undergoing constant acceleration. • Using this, we can figure out that the vertical displacement is 2.57 m 4) HORIZONTAL DISPLACEMENT • Horizontal displacement can be calculated using the above displacement formula, except that this time the time-up will include the whole jump, not just the ascension to the peak of the jump. • However, we must consider that the skater bends their knees a little when they land, so the time-down will include an additional 10 cm of flight. • This calculation is done in four steps: 1. Calculate time-up and jump height (same as above) 2. Calculate the total distance the skater will come down, adding the jump height to the difference in the height of ther center of mass from takeoff to landing (10 cm) 3. Use the displacement formula to solve for time-down, using the distance from the peak to the ground as D, where Vi = 0. 4. Get the total time, by adding time-up and time-down. • For more detailed instructions, visit: http://btc.montana.edu/olympics/physbio/biomechanics/pm11.html • In other sections (delete the last part of the URL) this site contains other information pertaining to the other topics covered in this glog, if you're interested 5) TAKE-OFF ANGLE • The final question is: What is the ideal take-off angle to get the biggest jump (and therefore the most time in the air to complete the revolutions? • First of all, the take-off angle is the angle formed from the horizontal line that stretches perpendicular from the centre of mass to the horizontal surface of the ice at the instant before the skater's bottom leg leaves the ice. • With an angle greater than 45 degrees, the jump has lots of height but little distance, • With an angle less than 45 degrees, the jump has lots of distance but little height • However, a jump that takes off a an angle of around 45 degrees will have good height and distance, meaning that the jump will last the longest in the air and allowing the skater to complete the most revolutions possible. • In conclusion, projectile momentum makes it all happen in a jump, since all the angular momentum in the world is useless if you can't get off the ground.
Projectile Motion
1) HOW TO LAND A TRIPLE AXEL • The principle of the law of conservation of momentum was used when a group of North American figure skaters were trying to figure out how to land a triple axel to keep up with the Russians. • Through careful video analysis, they found out that the skater stays in the air for the same amount of time as the single or double axel. • To get the extra half-revolution, the Russian figure skaters were getting into their tight jumping position more quickly, thus making the best possible use of their angular momentum; by reducing their inertia and increasing their angular velocity. 2) NEW FIGURE SKATE HAS ANKLE HINGE FOR SAFETY • Researchers at the University of Delaware have redesigned the figure skating boot. • The new boot is hinged at the ankle, and it has been proven through testing that the design lowers the stresses on the figure skater's body when he/she lands a jump. "The current boot is so rigid that it’s like putting your ankle in a cast. It forces skaters to land flat-footed, which leads to the injuries you see so often—sometimes to the foot itself but primarily to the joints," said Jim Richards, Distinguished Professor of Health, Nutrition and Exercise Sciences and director of the University of Delaware Biomechanics Laboratory. • The new boot allows the skater to land with their heel relatively high, allowing the landing impact to be absorbed much more effectively. • The new boot is scheduled to be released this summer. 3) NEW SYNTHETIC BOOT MATERIAL • Recently, figure skating boots have began to be made with synthetic materials and heat-moldable linings. • They combine strength with lighter weight than traditional leather boots, and are easier to break in. • On a side note (from personal experience), boots made out of synthetic materials tend to scuff much less easily and so stay new-looking for much longer than leather boots.
Inertia
