Physics of Skating

Wed. Nov. 27/13 Physics of Skating Celine Yau

Physics of SkatingFigure skaters and speed skaters have a basic understanding of physics. On ice, Newton’s

second law is so obvious; it is almost comical that it takes a few physics class to see the relationship

between pushing backwards and moving forwards. Most figure skaters do not comprehend why they

pull their arms in during a spin or jump, yet, they know by doing so, they will have more rotations in

a shorter time.Additionally, in jumps, skaters experience a change in energy causing their jumps to

have a parabolic path. Even the simple things, such as what angle to push off with has a technical

explanation.Figure skating and speed skating tie in so nicely because not only do both require ice but,

they both require the principal of centripetal acceleration. Unfortunately, the physics of skating does

not cover the artistic ability of figure skaters or the speed skaters’ passion, but it is a start to covering

this magnificent winter sport.

Sometimes, friction makes it harder to do certain jobs, however, in skating, without friction,

it would be impossible to move. When skaters move forward, they must push backwards with the

pushing foot, but what most people do not realize is that friction keeps the pushing foot from

slidingout. This form of friction is called static friction. The friction that allows skater to slow down

and stop is called kinetic friction. Skaters turn out their foot and push down against the ice to increase

the amount of friction between the ice and skate.

Furthermore, while most people believe that skaters glide simply because ice is slippery, the

truth is that friction and pressure also play a big role to allow the skater to glide. Skates are designed

so that all your body weight is supported by one -occasionally two- thin blades. Becausethere is a

high amount of pressure from your weight transferred onto the thin blade, the melting point of ice is

reduced. On top of that, the kinetic friction from the skate blade against the ice causes a thin layer of

water, thus it is not the ice that makes the skater glide so smoothly but actually the thin layer of water

on top of the ice. Arenas do not drop the temperatures way below 0⁰ because not only is it a waste of

energy but it would also be harder to produce a thin layer of water with excruciatingly low

temperatures.


Next, skating is an excellent way to explain Sir Isaac Newton’s laws of physics. To start off

simple, the easiest law to explain is Newton’s third

law: for every action there is an equal and opposite

reaction. When figure skaters wish to move forward,

they must push backwards against the ice. In the

diagram, it shows the skater is moving forwards by

pushing backwards and causing the ice to exert a force

(F) perpendicular to the blade. To jump, ice skaters

push down on the ice and the ice pushes back up (this

is also known as the normal force).

As for Newton’s first law: the law of inertia

“A body moving on a level surface will continue in the same direction at a constant speed unless

disturbed,” can be explained through spins or jumps. These two tricks have a few things in common;

the first thing is angular momentum. Several things determine angular momentum: the torque,

moment of inertia, and angular velocity. To understand angular velocity –also known as an omega

(ω)-, you must be familiar with angular displacement. Angular displacement is nearly the same as

linear displacement the difference is that angular displacement deals with angles. In other words,

The graph shows that around 0⁰C, there

needs to be around 1 P(atm) to change ice

into water. The P in the graph represents

atmospheric pressure which is the force per

unit area exerted by the air above it. This

explains why ice under more pressure will

melt more easily. Also, as the temperature

drops, more pressure is needed to convert

ice into water.

http://www.real-world-physics-problems.com/images/physics_ice_skating_8.png


angular displacement is the angle that a rotating body goes through; this is measured in degree or

radians.

Just like angular displacement is to linear displacement, angular velocity is the equivalent to

linear velocity. Angular velocity (ω) is the rate of change of angular displacement,to simplify: how

fast an object is rotating. Angular velocity is measured in radians/second or degrees/seconds.

The next component to angular momentum is the moment of inertia.As explained by

Newton’s first law, an object at rest stays at rest and an object in motion stays in motion unless an

outside force is applied. Once that law is explained, the name for moment of inertia is quite simple;

moment of inertia is the measure of an object’s resistance to changes in its rotation (it is measured in

m4 or kbm2). When the mass is spread out, it is less likely to tip over; the opposite is true for when

the mass is closer to the center. For example, tight rope walkers hold onto a pole (horizontally), this

helps them maintain balance and not fall over as easily.

In this picture, the arrow started on the x-axis and

moved to red line, then the purple line. Just like linear

displacement, a direction must be shown. For this case

we can choose counter clockwise, this makes counter

clockwise motion positive and clockwise motion

negative. The displacement can be expressed with the

equationθ=θ1+(−θ2).

In this case, to determine the angular velocity:

ω=(θ2−θ1)/∆ t


To review, the chart below shows the connections between translational quantities to the

rotational equivalent mentioned in this report:

Translational Quantities Rotational Equivalent

Force F (N)Mass m (kg)Time t (s)Displacement d (m)Velocity v (m/s2)

Torque I (N∙m)Moment of Inertia I (kg m2)Time t (s)Angular displacement θ (rad)Angular Velocity ω (rad/s 2)

Back to the original topic, angular momentum (H) measures

the tendency of an object to keep moving.The equation for angular

momentum is H=Iω where Irepresents the moment of inertia and ω

is angular velocity. Similar to energy, angular momentum is

conserved unless an outer force acts upon it. When figure skaters start

spinning, they spread out their arms and extend their legs; this causes

them to have a higher moment of inertia because their mass is further

away from their axis of

rotation –the centre of where a

skater rotates.

To spin faster, figure skaters

bring their arms into their axis

of rotation; this will cause

their moment of inertia to

lower. Since angular

momentum is conserved,

naturally, their angular

velocity will increase, meaning the figure skater will have more rotations in a shorter amount of time.

There are two ways a skater can stop: by increasing their moment of inertia or waiting for

friction to do it for them. By increasing the moment of inertia, the angular velocity lowers because

http://www.einstein-online.info/spotlights/angular_momentum

http://www.freewebs.com/ayxl/figureskate.jpg


(Iω)=(Iω)1. Next, do recall how angular momentum is conserved unless an external force acts upon

the object. For figure skating, the only external forces are gravity and friction. Since gravity acts

vertically through the skater’s centre of mass, it does not change the skater’s angular momentum. So,

only when the force of friction overcomes the force applied will the skater slow down and come to a

stop.

There are several parts to generating angular momentum for spinning on ice. The first part

would be applying a force against the ice then -as mentioned earlier-, the ice would push back against

the skater and propel them forwards. This force must be applied a certain distance away from the

axis of rotation. If the force is applied directly through the skater’s axis of rotation, the skater will not

spin. In addition to that factor, the force must cause a torque.Another way to look at torque is that it

is like a force (push/pull in a specific direction that moves an object), except it is applied to gain

angular momentum. A torque –which is measured in N∙m-, is the tendency of a force to rotate an

object around an axis, it can be thought of as a twist to an object. To find torque use the equation:

T=r × F where r is the position vector starting from where the torque is to the point of the applied

force; F is the force. Because torque is dealing with the multiplication of vectors, × can be used. In

terms of spinning, × can be substituted withsin θ, therefore the equation becomesT=rF sin θ.

Figure skaters want to start off with a large moment of inertia so that later on, they can

increase their angular velocity. Keep in mind, with a larger

moment of inertia, the more force it takes to start

spinning. If there is a larger F value, and same r value,

naturally, the product (torque) will be larger. As a result

of the torque being larger, the angular momentum will be

greater too, allowing the skater to spin faster.

Another major component to figure skating is the

jumps. Figure skaters want their jumps to be both high and long. The key factors that come into play

when trying to achieve a high and long jump are take-off angle, take-off velocity and take-off height.

http://zonalandeducation.com/mstm/physics/mechanics/forces/torque/torque2.gif


If figure skaters take off at around 80⁰there jumps will be very high but not very long. Also, when

figure skaters take off at around 20⁰ there jumps will be longer but not high. Ultimately, to have a

nice balance of height and length, the take-off angle should be around 45⁰; this allows the skater

enough height and time to finish all their rotations in the air.

When people jump forwards, their path is

in a parabolic shape. This is because the gravity

is constantly causing an acceleration of

approximately 9.8m/s2 [down]. When skaters

push off the ice, they have Kinetic Energy –which

is energy due to motion. Then, as the skater increases their height, the Kinetic Energy is converted

into Gravitational Potential Energy –which is energy an object possesses because of its position in

the gravitational field.When all the Kinetic Energy (Ek) has transformed into Gravitation Potential

Energy (Eg), this is the max height that the skater will reach and for a millisecond, the skater’s

velocity will be 0. Lastly, gravity will accelerate the skater’s velocity back down to the ice.

“What goes up must come down.” –Isaac Newton

As for the horizontal displacement, according to Newton’s third law, when the skater jumps,

he/she must push down and backwards on the ice. However, in reality, the skater has already

converted chemical potential energy (muscles power) into kinetic energy, thus the skater already has

quite a bit of horizontal velocity. So in actuality, the skater mainly needs to focus on pushing down.

Once the figure skater has hit the ground, the gravitational potential energy they had at the peak of

the jump is converted into kinetic energy.

Just like spinning, the skater needs to start with a high moment of inertia so that they can

increase their angular velocity. After take-off, skaters cannot change their angular momentum

because the only external force acting on the skater is air resistance. However, due to the small

amount of height (relative to the Earth’s atmosphere) that the skater reaches, air resistance does not

affect the skater’s angular momentum.Since there is only so much a person can spread out their mass,

http://farm3.staticflickr.com/2769/4215796779_a80fb5c170.jpg


the more complex the jumps become (doubles, triples, and quadruples spins in the air) the tighter the

skater must squeeze to yield enough angular velocity.

Moving away fromfigure skating, the next progression for the physics of skating is speed

skating. Speed skating seems quite simple, but just like every sport, there is a technique that goes into

the exercise. When skaters push back with their foot, they don’t actually push off at perpendicular

angle to the blade. They push back 35° -as illustrated in figure (b)- from the blade. This allows 77%

of the force to be transferred into movement compared to the 70% energy transfer if they pushed

perpendicular to the blade (illustrated in figure (a)).

The max speed that a skater can reach is

influenced by how fast the skater can move their legs.

The average maximumhumanspeed for runners is 43

kph. However, skaters can reach up to 50kph. The

reason for this is because the 43kph is relative to the

ground. While skating, the skate on the ice is gliding

forwards while the skater is moving their other foot

forwards. This means that during one step the skater’s

gliding foot moves about 0.7v because:Gliding for Gold: The Physics of Winter Sports

By usingcosθ=ah

, you can find the speed of the

skater relative to the direction they are going.


The next stride allows the skater to move back towards

the way they want to go thus, the next stride contributes

to the skater’s speed. Therefore the total speed will be

1.4v. And if the max leg speed is around 43 kph, then

skaters’ theoretical max speed will be 1.4 ( 43 )=60.2 (not

including friction).

Once the skater reaches the maximum speed, the

skater does not need to overcome friction or drag, and there is no need for the skater to increase

speed thus, less energy is needed.On top of that, at maximum speed, it becomes harder for the skater

to push against the ice. When the skater’s speed reaches the speed of his/her legs, it results in no

force exerted on the ice.

When watching speed skating, you will notice how skaters will lean towards the centre of the

rink. They must do this because of centripetal acceleration (ac).Centripetal acceleration is the rate of

change of tangential velocity.Tangential velocity is the instantaneous linear velocity of an object

moving in a circular path, and its direction is tangential to the circle at that point. At the ends of the

ice rinks, speed skaters must travel in a semicircle, during that time, they are travelling in uniform

circular motion. Because the direction of velocity is always changing, there is always acceleration

during uniform circular motion. This acceleration is pointed towards the center of rotation, and is

where centripetal (“center seeking”) acceleration receives its name from. To calculate the direction of

the change in velocity, two directions of the velocity of an object are picked and inputted into

ac=∆ v /∆ t.

Next, to calculate the magnitude:triangles ABC and

PQR are similar triangles thus∆ vv1∨2

=∆ sr and is rearranged to

find Δv: ∆ v= vr

∆ s. We divide the equation above by the Δt

http://cnx.org/content/m42084/latest/


producing: ∆ v∆ t

= vr

×∆ s∆ t

. That is the same as ac=v2

rbecause centripetal acceleration is

∆ v∆ t

. This

equation can become more simplified into

ac=(rω)2

r=r ω2.

Fy is 0 because there is no vertical

acceleration. To deal with the horizontal

force, we combine the equation for centripetal

acceleration and Newton’s second law

yieldingF x=mv2

R (equation 1). Because θ is

constant, this means the skater is in a state of

rotational equilibrium –which means the total angular acceleration is 0.To show this mathematically:

F x sinθ ∙ L−F y cosθ ∙ L=0 (equation 2).Onceequation 1, equation 2 and centripetal acceleration are

combined, it produces the equation tanθ=Rg

v2 and can be used to calculate how much a speed skater

must lean so that they can turn around the rink. For example, if a skater going 20m/s around a

semicircle with a radius of 8.5m, the skater has a lean of 11.8°.

Key for the photo:

g is the acceleration due to gravity, which is equal

to 9.8 m/s2 on earth.

G is the center of mass of the system (including

the skater’s skates).

P is the approximate contact point between the

skater's blades and the ice.

L is the distance between point P and point G.

Fxis the horizontal contact force, with the ice,

acting on the skater's blade at point P.

Fy is the vertical contact force with the ice acting

on the skater's blade at point P.

R is the radius of the turn, measured from the

center of the turn to the center of mass.

ac is the centripetal acceleration of point G.

θ is the angle between the horizontal and the line

passing through points P and G. This measures

the “lean” that the speed skater has.


By understanding centripetal acceleration, we can move back into figure skating and more

specifically, the death spiral (shown in the image below) presented in pairs figure skating.

This famous position deals with

centripetal and centrifugal force. The man in the

picture pulls the woman into the pivot point (P);

he acts like the centripetal force in this system.

Centrifugal force comes from the words centrum

–which means “center”- and “fugere” - “to flee”.

In this system, the woman acts like centrifugal

force. However, in reality, the performers will

skate into this position and thus the only reason

the woman is pulling away is because of her

inertia.

For the two skaters to maintain rotating around a point on ice, the man must use one foot to

push against the ice. A free body diagram that shows the skaters as one body (as drawn bellow) helps

determine the amount of force that the man must exert on the ice.

ac is the centripetal acceleration of the center of

mass M due to the rotation rate v. As discussed in the

speed skating section, centripetal acceleration points

towards the center of the circle.

FP is the force exerted by the ice on the male skater's

blade at point P. This force acts in the direction of the

centripetal acceleration ac.

Then, because of Newton’s second rule,∑ F=ma,Fp can be seen as F p=m ac. Since centripetal

acceleration isac=ω2 R, the equation becomesF p=(mA +mB)ω2 R. To put this equation to use, the

women has an approximate mass of 50 kg (mA = 50 kg) and the man has 80 kg (mB = 80 kg), ω is 308


rad/s, and R is 0.4 m. As a result,Fp equals 750 N indicating the man must push down with a force of

750N to stay in place.

Skating has many different things happening at the same time; even the most beginner thing

like taking a “step” forward causes melting, friction, and proves one of the three most important laws

of physics. Athletes who skate at home have a greater advantage because they know the temperature

and the amount of pressure they must apply to melt the ice.Figure skating isan excellent example of

angular acceleration and torque; by watching the Olympics,these different elements can be spotted in

the figure skaters’ spins.With the understandingof angular acceleration, comes the knowledge of

understanding angular velocity, and moment of inertia.The greatest thing about understanding energy

transfer from kinetic to potential is that while watching figure skaters, you realizethe amount offorce

that a skater must overcome during their jumps. This really puts some perspective inwatching the

athletes perform. Once again, angular momentum is another physics element that is incorporated in

jumps; the more complex the jumps are, the tighter the skater must pull in.

Once you know how to skate, it iseasier than running because at maximum speed, the skater

does not need to use as much force to move forward.At maximum speed,the skater does not need to

overcome friction/drag. The complexity of speed skating comes in when skaters need to turn around

at the end of the rink. Due to centripetal acceleration, skaters must lean a certain degree in order to

complete the turn; this “lean” depends on the radius of the semicircle, and the speed.

Moving back into figure skating, during a death spiral, the couple experiences a centripetal

force. So that the pair continues to turn around on an axis, the man must apply a force against the ice

with one foot.The pressure of this force is based on how much the couple weighs, how big their

circle is, and how fast their angular velocity is. Understanding the physics to speed skating and figure

skating can make it easier to fix the small errors in skating techniques. Still, just like every other

sport, there is a reason why the nerds (despite their knowledge of physics) still do not fare so well in

competitive skating. The jumps, spins, and the speed all require practice, plus some cuts and bruises

to achieve.


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Physics of Skating