An evaluation on the physics of the flying disk

1

An Evaluation on the Physics of the Flying Disk

Ishan Amin

An evaluation of the Physics of the Flying Disk 2

Ishan Amin 664854

Contents Introduction: ................................................................................................................................................. 3

History: .......................................................................................................................................................... 3

How does the Frisbee Fly? ............................................................................................................................ 4

Aerodynamic Forces: .................................................................................................................................... 6

Mathematical Description: ........................................................................................................................... 7

Gyroscopic Properties: ................................................................................................................................ 10

The Effect of Turbulence: ............................................................................................................................ 12

Trailing Vortices: ......................................................................................................................................... 14

Appendix ..................................................................................................................................................... 15


Ishan Amin 664854

Introduction:

The Flying disc or Frisbee, as it is more commonly referred to as, is a circular piece of plastic, which

produces greater revenue than footballs, baseballs, and basketballs combined. At first, a Frisbee was

nothing more than a pie tin that college students threw, today, however, this flying toy has been the

basis for over 20 sports, including Ultimate Disc, Frisbee Golf, and more. These sports are played at both

an international level as well as a regional level.

This circularly shaped piece of plastic consists of a concave and convex side and is being thrown at parks,

beaches, etc. as we speak. This toy is used by millions around the world, yet not many know how this

device works, and what it is capable of doing. This product is utilized so often that many tend to

overlook the brilliance, simplicity, and ingenuity of the product.

I have comprised this project to explain to others, as well as learn, how the flying disk works, and what it

is capable of doing. These goals will be accomplished by the production of a “Trick Shots Video”, to show

the capability of what the flying disk is capable of doing, and a report on the physics of the disc,

demonstrating how the flying disc works.

History:

The invention of the Frisbee, surprisingly, was inspired by pie tins. In 1871, William Russell Frisbee

purchased a bakery in Bridgeport, Connecticut. He named this company the Frisbee Pie Company. Since

it was close to Yale University, the majority of William’s customers were university students. After

completing the pie, the students realized that the pie tins achieved flight when thrown in the air. This

became a popular pass-time amongst the university students. Because the name “Frisbee” was printed

on the pie tins, the students started to call the pie tin a “Frisbee.”

The Discuss is a flat circular disc made of iron. It was first introduced, and utilized, at the Olympics,

dating back to 5 B.C. It is now used in Track-and-Field competitions, the Olympics, Pan-Am, etc. The

Discuss was the first flying disk to be utilized and introduced at a sporting event. It has been rumored

that a Yale University student was studying ancient Greece culture, which led him to the Olympics, and

then to the Discuss. He noticed that the Frisbee pie tin and the discus shared similar qualities, and the

rest is history.

In 1948, Fred Morrison discovered this popular pass-time amongst the university students. Being the

part-time inventor that he was, he created plastic and more aerodynamic version of the pie tins. Soon

after he patented the product and began to sell it at county fairs, amazing audiences everywhere. In

1955, Fred Morrison sold the patent and rights to the Frisbee to a Californian toy company called

Wham-O. After two years the company began to mass-produce the product, officially trademarking

Frisbee as the name of the product. This was a huge success, selling over 100,000,000 units. Mattel, the

toy company, now owns the rights to the Frisbee. There are also 60 other businesses that market their

versions of the Frisbee.


Ishan Amin 664854

How does the Frisbee Fly?

Many objects that are created to achieve lift for long periods of times consist of engines or motors that

assist flight, lift-off, and landing. The Frisbee is unique in the sense that it can achieve lift without the

use of motors or engines. All that is required is the initial lift-off.

There are many factors that affect the ability of the Frisbee to achieve lift. However, most of these

factors affect the Frisbee in such a minuscule manner, that they can be completely ignored. The two

most important factors are Gyroscopic Inertia or Stability and Aerodynamic Lift.

Shape of the Flying Disc: Before we begin to talk about the physical concepts of the disc, let’s discuss the actual physical disc. The

Disc is a circular shaped piece of plastic, usually 175 grams, and 17 cm in diameter. There are two main

keep components in this circular shape, the first is the thick rim, and the second is the ring of grooves

located along the perimeter of the disk.

Rims: The thick rim, making up the edge of the Frisbee, plays many important roles in the mechanics and

motion of the Frisbee. Frist and foremost, the rim allows the user to both grip and throw the disk with

more ease.

Moment of Inertia, I = MR2 [I represents moment of inertia, M represents mass, r represents the distance

between axis and rotation mass]. Inertia is a property of matter that states that all matter remains in the

same state, and continues moving in a straight line (linear motion) until an external force changes the

state or linear motion. Therefore, the moment of inertia, or the rotational inertia, is a quantity that

expresses a body’s tendency to resist angular acceleration. A number that can identify how easily the

body can change its direction while rotating.

The thick rim increases the stability of the disc by increasing the moment of inertia of the disc at about

the axis of symmetry, as can be seen by the graphical depiction of a Frisbee’s flight below. The rim can

increase the moment of inertia due to the fact it provides a curvature to the edge of the disk that

creates the desirable airfoil shape that will be discussed further.

The effect of the rims can be viewed in the graph below, depicting the flight of a Frisbee. Unlike a ball,

which would create a parabola (if graphed), the disc does not begin to descend when it reaches the

maximum height. If the disk were flat, it would have a lower ascension than a Frisbee with rims, as well

as a quicker and more abrupt dissension.


Ishan Amin 664854

“Finally, the cupped region on the bottom of the disc substantially increases the coefficient of drag from

the vertical profile, while the horizontal profile is still somewhat streamlined. As a result, as the disc

begins to fall, the cupped profile behaves like a parachute, and the horizontal component of drag dwarfs

the vertical component. This allows the flying disc to be thrown much further than a ball of equivalent

velocity.” (Anthony Scodary, 2007).

Grooves: The Kutta Effect: the effect in which surface harbors a jet flow that remains attached to it, even when

the surface diverges from the initial direction of the jet flow

It is not is not uncommon to find concentric grooves located on the top surface of a flying disc. The

grooves act to trigger turbulence at the leading edge. The same principal is shown with a gurney flap on

an airfoil. The turbulent region keeps the flow of air above the disc attached to the disc, increasing the

lift. This allows the disc to be thrown at a greater angle of attack before it stalls. A stall is created when

the flow of air separates itself from the disc causing a decrease in the lift as well as drag.


Ishan Amin 664854

Aerodynamic Forces:

Aerodynamics is the study of how air interacts with solid bodies that move through the air. In

aerodynamics, there are four major forces of aerodynamics: Lift, Thrust, Drag, and Weight. All of these

forces have an effect on the solid body that is moving through the air. The four forces determine the

direction in which the object is moving, and the speed of the object. An Aerodynamic object is one that

is shaped in a way that allows it to reduce the drag from the air moving past. To determine if an object is

more aerodynamic than another, both the objects have to have the same height at which it is

dropped/thrown, and the same amount of thrust. The one that drops the fastest, or flies the furthest is

the more aerodynamic object.

All of the four forces, Lift, Drag, Weight, and Thrust, effect the flight of the Frisbee, but lift and drag are

the two main aerodynamic forces that act upon the Frisbee.

Lift: Lift is created by the a solid object moving through a fluid, and turning a flow of air in the process. It

directly opposes the force of weight; this implies that lift pushes an object upwards. Lift acts through the

center of pressure of the object. Since lift is classified as a force, it falls under a vector quantity.

Therefore, it has both magnitude and direction. The direction of the lift is perpendicular to the flow of

direction. The three main factors that affect lift are, air (mass, viscosity, and compressibility), object

(shape and size) and motion(velocity and inclination to flow).

Lift can be calculated by using the following formula:

𝐿 = 𝐶𝑙 ∗𝜌∗𝑉2

2∗ 𝐴

Lift is equal to coefficient multiplied by density(𝜌) times velocity squared over two, multipled by the

wing area.

*Note: The coefficient (𝐶𝑙) is comprised of all complex dependencies and is determinesd expirementally.

Dependencies can include wind patterns, alttitude, etc.

The generation of lift can be explained using two principles/laws of physics. Bernoulli’s Principal and

Newton’s Third Law. These are two ways of looking at the same thing and are both considered valid

reasonings.

*Note that for an object to achieve lift, it has to be able to have an angle of attack that is greater than

0°, or have an asymmetric or cambered shape.

Bernoulli’s Principal states:

“As the velocity of a fluid increases, the pressure exerted by that fluid decreases.”

In tactical term, Bernoulli’s Principal states that the faster the air, the less pressure it exerts on whatever

it is flowing over. As the air moves over the disc, it speeds up, thus creating lower pressure on the top of

the disc. Similarly, the air at the bottom has to move around the edges of the rim, which slows it done


Ishan Amin 664854

and allows it exert a higher pressure. This higher pressure pushes the Frisbee up, more than the lower

pressure pushes the Frisbee down, and thus creates lift.

Newton’s third law of motion states:

“To every action, there is always opposed an equal reaction: or, the mutual actions of two bodies upon

each other are always equal and directed to contrary parts.”

As mentioned before, the object in motion has to have an angle of attack that is greater than 0° to

achieve lift (if the angle of attack is 0°, the object will achieve flight. However it will continue in a linear

motion, and not go higher, that is to say, achieve lift), this means that the front of the Frisbee is pointed

upwards. At 𝑎 > 0°, the Frisbee will be in a slant which will cause the wing to deflect and slow the air

down. Since the air is being pushed back and down, it will push the wing up and back. The air pushing

the wing up, creates the lift and the air pushing the wing back, creates drag.

Typically, a Frisbee is designed in such a way that that the Center of Lift, is in front of the center of mass.

This is done to ensure that the disc has a neutral angle of attack, or tilt while being integrated into the

airflow. However, this does have a disadvantage, in which the disc will favor one side more than the

other, and tilt towards that particular side. This can be easily resolved by introducing a spin, or angular

velocity. This allows that tilt to transform into a pression rate.

Drag: Drag is the force that slows a moving object. It resists the moving object. The objects shape usually has

an effect on the drag, for example, round surfaces have less drag than flat surfaces, and narrow surfaces

have less drag than broad surfaces. An example of this is when someone sticks their hand out of the car.

When the hand is horizontal, there is less drag, and when the hand is vertical, there is a greater drag.

The more aerodynamic the shape, the smaller the effect of drag will be placed upon it. Drag can be

described as an aerodynamic friction. One of the sources for the cause of drag is the friction between

the air molecules and the surface of the object in motion; this is referred to as skin friction. A smooth

surface object produces less friction than a surface with a rougher edge. Similarly, the more viscous the

air, the more drag it will produce, and vice versa.

Mathematical Description:

To calculate the drag on a Frisbee, we must first find the Reynolds number, so we know which drag

relation applies to the Frisbee.

𝑅𝑒 =𝑝𝑣𝑑

𝑛,

The 𝑝 is the density of the fluid (air), 𝑣 stands for the velocity of the fluid, d represents the diameter of

the Frisbee, and the n is the viscosity of the fluid.

*Note: The equation above is a simplified version of the Reynolds Number.


Ishan Amin 664854

𝑅𝑒 = 1.225 ∗ 14 ∗ 0.260

1.73 ∗ 10−5

=4.459

0.0000173

= 257745.6647

For a Reynolds number of this size, one must now calculate the PrandtI relationship to further this

calculation and calculate 𝐹𝑑, the drag:

𝐹𝑑 = −𝐶𝐷𝜌𝜋𝑟2𝑣2

2= −

𝐶𝐷𝜌𝐴𝑣2

2

𝐶𝐷 is the drag coefficient that varies with the object, and is given as a quadratic function, and is only

dependant on the angle of attack. The angle of attack, which is given by 𝛼, is an angle that is formed

between the relative velocity vector and the plane of the Frisbee. The angle of attack is the angle that

the Frisbee is thrown at. A positive angle of attack has an angle is above 0°, the launch angle, while a

negative angle has an angle less than 0°.

𝐶𝐷 = 𝐶𝐷0 + 𝐶𝐷𝛼(𝛼 − 𝛼0)2

𝐶𝐷0, 𝐶𝐷𝛼, and 𝛼0are constants and vary based on the physical features of the Frisbee.

The lift force that acts on the wings of an airplane and the force of lift that acts on a Frisbee are similar.

Both are calculated by using the Bernoulli’s principal. In fluid dynamics, the Bernoulli’s principle states

that fluids that flow fast velocity have low pressure while fluids with a slower velocity have a higher

pressure.

Mathematically written as Bernoulli’s Equation

𝑣21

2+

𝑝1

𝜌+ 𝑔ℎ1 =

𝑣22

2+

𝑝2

𝜌+ 𝑔ℎ2

𝑣 is the velocity of the fluid, 𝑝 stands for the pressure of the fluid, 𝜌 is the density of the fluid, 𝑔 is the

acceleration of gravity and ℎ is the height of the fluid. “1 and 2 refer to different points in the fluid along

the same streamline.” (Hummel 2003). For the sake of simplifying the equation we will assume that the

height difference between the air flowing above the Frisbee and below the Frisbee is insignificant, in

result cancelling out both the height variables. We will also have to explore the possibility that the

velocity of the air flowing above the Frisbee is the same as the air flowing underneath, due to the fact

that the difference in the path length is constant. In other words: 𝑣1 = 𝐶𝑣2. Therefore our new equation

is:

For a standard Frisbee thrown at sea level, the density of

air is approximately 1.23 kg/m3, the velocity of an

average Frisbee throw is 14 m/s. The diameter of a

standard Frisbee (accepted by the National Ultimate

Association) is 0.260 m and the viscosity of air is

1.73×10−5 Ns/m2. This gives Re equal to 2.58 × 10^5.”

(V.R Morrison).


Ishan Amin 664854

𝐹𝐿/𝐴 = 𝑝1 − 𝑝2. 𝐹𝐿 is the lift force, while the 𝐴 stands for the area of the Frisbee. To solve for 𝐹𝐿 the

following equation is used:

𝐹𝐿 =1

2𝑝𝑣2𝐴𝐶𝐿

To determine the equation above, the 𝐶 coefficient is incorporated into the 𝐶𝐿 coefficient. The

coefficient is a linear function of the angle of attack.

𝐶𝐿 = 𝐶𝐿0 + 𝐶𝐿𝛼𝛼

[The Cl0 and Cla are constants and both dependent on the physical properties of the Frisbee itself]

Gravity:

The force of gravity applies to all objects on Earth, stationary and in motion. The force of gravity, on a

Frisbee, acts on its center of gravity to bring it towards the ground. Gravity [on Earth] is a uniform and

constant vertical force; that isn’t dependent on the angle of attack or speed of the Frisbee. The force of

gravity on Earth is 9.807m/s. Gravity acts on the center of mass of the Frisbee to bring it down. The

center of mass is a point of the Frisbee, where if a force is applied, the Frisbee will accelerate towards

the direction of the force. The motion of the Frisbee would be classified as translation motion, linear

motion. This is also called the center of gravity.

Airfoil:

An airfoil is a structure with a curved surface, usually one concave and one convex, which is expressly

designed to produce a desirable ratio of lift and drag while in flight (motion). Airfoils are used as a basic

form of wings, fins, and horizontal stabilizers in many aircraft.

A Frisbee is identified as an airfoil, as it does have both a concave and convex surface. Additionally, it is

designed to generate more lift than drag, much like the wings of an airplane.


Ishan Amin 664854

Being an airfoil, the disc will experience gravitational and aerodynamic forces acting upon to accelerate

the disc according to Newton’s Law:

𝐹 = 𝑚𝑎

These forces will have a varying effect on the motion of the disk, for a multitude of reasons. Some of the

reasons include the angle of attack, the angle between the plane of the Frisbee and the relative velocity

vector, etc.

Gyroscopic Properties:

A gyroscope is defined as a device that consists of a wheel or disk mounted so that it can spin rapidly

about an axis that is itself free to alter in direction. The orientation of the axis is not affected by tilting of

the mounting; allowing gyroscopes to provide stability as well as maintaining a reference direction in

navigation systems, automatic pilots, and stabilizers.

Two properties that all gyroscopes share are Rigidity and Precession:

Rigidity – The spin axis, also known as the axis of rotation, of the gyro wheel, must retain its direction in

space if no force is applied. The axis of rotation, the center point, of the disc remains in the fixed

direction. This holds true for the Frisbee.

Precession – The spin axis tends to push back at a right angle to the direction in which the force is

applied. Both of these properties apply to a disc. Precession is witnesses when individuals perform the

bounce shot, in which the individual releases the disc towards the ground, and has it bounce back at a

90° angle. Therefore, deeming it a gyroscope.

The rotation of the Frisbee is an important component of the mechanics of the Frisbee. The rotation of

the disc is what allows it to be able to achieve a long-lasting, stable flight. Without the rotation the

Frisbee would simply fall a couple meters in the vicinity to its initial distance, depending on the force

applied. Notably, as mentioned before, the lift on the front of the Frisbee is slightly larger than it is on

the rear. This discrepancy creates a torque on the Frisbee, and when the Frisbee is not spinning the little

torque flips the front of the disc up.

Torque is a twisting force that has a tendency to rotate an object about an axis, fulcrum, or pivot. Lift

and drag are classified as a push or pull forces; similarly, torque is classified as a twisting force.


Ishan Amin 664854

Inertia is a property of mass that states that mass has a tendency to remain in constant motion, or at

rest. If the mass is moving, inertia states that it should travel at a constant speed in the same direction.

Gyroscopic Inertia refers to the resistance one encounters once making an attempt to alter the axis of

rotation of a rotating object. In short, gyroscopic inertia is nothing more than the magnitude of the

angular momentum.

When a Frisbee is thrown with a large spin, it possesses large amounts of angular momentum, which has

a vector that is either in the negative or positive vertical direction. Keep in mind that due to the fact that

the Frisbee is an extended object (an object that spins on an axis [Ex: Gyroscope]) rather than a point

object (an object that orbits a point [Earth and Sun]), we use the equation:

𝐿 = 𝐼𝜔,

Where the I represents rotational inertia while the w represents the angular velocity, the speed of an

object traveling rotationally.

Proof of Concept:

To calculate the angular momentum of a point object the equation is as follows,

𝐿 = 𝑟 ∗ 𝑝,

Where r represents the radius and the p accounts for the linear momentum.

A Frisbee is a solid object that rotates about the origin. Assign the angular velocity of the object 𝑤. The

radius of the Frisbee, 𝑟, is moving at the velocity of,

𝑣 = 𝜔 ∗ 𝑥

A differential mass 𝑑𝑚 of the body therefore contributes a differential angular momentum,

𝑑𝐿 = 𝑑𝑚 ∗ 𝑟(𝑤 ∗ 𝑟)

As provided by Eugene Motoyama, 𝑑𝑚 is divided to the other side, and we have the following equation,

𝑟(𝑤 ∗ 𝑟)

The equation provided above is then integrated into the equation prior to it, which translates to

𝐿 = 𝐼 ∗ 𝜔

The I, as mentioned above, represents rotational inertia. This rotational inertia is calculated by the

following equation,

𝐼 = 𝑀𝑅2


Ishan Amin 664854

Where M is the mass and R is the radius. The mass of a regular Frisbee is 175g while the radius is

0.130m, both these facts have been approved by the Ultimate National Association. Therefore, we can

assume that a Frisbee would have a consistent rotational inertia of:

𝐼 = (0.175𝑘𝑔)(0.13𝑚)2

= (0.175𝑘𝑔)(0.0169𝑚2)

= 2.9575 ∗ 10−3𝑘𝑔/𝑚2

Angular velocity, 𝜔, cannot be calculated as it depends on the force at which the Frisbee is thrown at,

and the external conditions.

Torque Torque, as we have mentioned before, is a twisting force that tends to cause rotation. Every object has a

center of gravity/mass, that, if pushed would accelerate the object in that direction, this would be

translational motion. However, what if the object is pushed from another point, we get both a rotational

and translational motion. The equation for torque is,

𝜏 = 𝑟 ⊥ 𝐹

Where r is the radius, which is perpendicular to the axis of rotation, multiplied by force, which is

perpendicular to the radius.

The effect that the torque has on the body is the time rate of change of the angular momentum.

Therefore, we can write the equation as so,

𝜏 =𝑑𝐿

𝑑𝑡

We can further simplify the equation using the product rule. Since 𝐿 = 𝐼𝜔, torque can be written as,

𝜏 = 𝐼 ⋅𝑑𝜔

𝑑𝑟+

𝑑𝐼

𝑑𝑡⋅ 𝑤

By applying torque, 𝑤 (angular velocity,) is subject to change, as is the rotational inertia. The first term

of the equation corresponds to torque changing the angular velocity. If the object is stationary, and a

torque is applied, the object will start to spin along the torque vector. If the object is already in motion,

then the object will spin up or down. The second term signals a change in the orientation of the object.

The Effect of Turbulence:

There are two different types of flows, Laminar, and Turbulent. Laminar flow is a constant stream while

a turbulent flow is a laminar flow which accelerated and, due to this high speed, creating vortices.

As mentioned before, the cross-section of a Frisbee and Airplane wing share many similarities. However,

there is one major difference between the two; while the airplane wing is streamlined, the Frisbee is

not. Which means that the airplane wing is designed to present little resistance to a flow of fluid,

increasing speed, and ease of movement. Therefore, a turbulent flow affects a Frisbee far more than it

does to an airplane wing.


Ishan Amin 664854

A Frisbee and airplane have the same “head” shape, both of the heads are curved. However, in a

streamlined object, the tail is known to rescind while the tail of the Frisbee curves away. The rescinding

tail is important as it is the region where the momentum from the outside flow is transferred to the

boundary layer.

Air flows in layers, and when an object is in motion, the layer directly underneath and above the object

stick to the object. The flows are then accelerated by the movement of the object, resisting the motion

and creating drag. This chain continues, as the first most layer of air accelerates the second most layer

of air, and so forth. Eventually, the velocity becomes so minuscule that it can be considered as 0. This is

called the velocity gradient. Similarly, other physical properties such as pressure, viscosity, etc. also

possess gradients.

In a streamlined object, the velocity at the front and back of the object is 0, meaning that the maximum

velocity is between the front and the back. According to the Bernoulli’s Principal, this means that the

highest pressure is at the head and the tail while the lowest is in between the two. However, a thin

viscous boundary layer at the surface does drop the velocity to 0 and is the only force that produces any

drag. Due to the fact that the pressure is equal at the front and the back, there is little pressure

contribution to drag.

A blunt object (a Frisbee, one that is not streamlined) creates Turbulence through boundary layer

separation. The boundary layer is being acted upon by many different forces, the most prominent being

the pressure gradient, as the fluid particles simultaneously gain and lose momentum. The loss of

momentum is from the viscosity gradient while the increase is from the pressure gradient. This gain and

loss occur when the fluid is traveling from the point of maximum pressure to the point of minimum

pressure.

During the latter portion of the movement, the pressure gradient is reversed, as the Frisbee is slowing

down, and the boundary layer travels to the point of minimum pressure to the point of maximum

pressure, at the tail. For a streamlined object, this would not be possible without the gradient system, as

the outside layers assist the boundary layer to reach the tail, by increasing its momentum. However, the

outside layers that surround the boundary layer of a blunt object do not have the necessary momentum

to assist the boundary layer to reach the tail. The boundary layer then loses all of its momentum and

stalls. This causes the boundary layer to move in the opposite direction and separate from the object.

The overall result is a large turbulent wake. This wave increases the drag on the object, thus slowing it

down. In conclusion, a faster fluid speed causes an early boundary separation, which leads to turbulence

and drag.

As the speed of the fluid increases, the drag upon the object decreases. This occurs due to the fact that

the boundary flows transitions from a laminar flow to turbulent flow. The turbulent flow allows the

boundary layer to mix with the outside layer. Therefore, the outside layer contributes more momentum

to the boundary layer and delays boundary separation. “The decrease in pressure drag from the smaller

wake mire than makes up for the increased viscous drag.” (Eugene Motoyama)

Turbulence also occurs under the cavity of the Frisbee, the boundary flow detaches itself from the

Frisbee for a short period of time and reattaches to hit the trailing underside of the Frisbee. In addition

to creating a turbulent flow, this creates a small flow that resists the direction of the general airflow,

thus adding to the drag.


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Trailing Vortices:

Vortices are masses of swirling fluids that are formed by repetitive motion. Trailing vortices are an

inevitable product of a finite wing span. This phenomenon occurs when the air from the top of the

Frisbee and the bottom intersect, causing a turbulent wake. This wake, if the angle of attack is high

enough can cause vortices.

Vortices are created by airplane wings as well and appear as white tornadoes behind the wing. Many

assume that these vortices are exhaust from the engines, due to the color, however, the white color is

often due to the water molecules at that altitude.

The two vortices are formed within a close vicinity of each other, and due to the fact that both spin

inwards it thrusts the Frisbee, ever so slightly.


Ishan Amin 664854

Appendix

Angle of Attack: The angle of attack, represented by 𝑎, is the angle at which the Frisbee is launched at in relation to 0

degrees.

The Reynolds Number The Reynolds number was created by Osborne Reynolds, who discovered that a combination of

variables could be utilized to distinguish flowing air into two types: Laminar and Turbulent. Using

Dimensional analysis, you could find relationships in length, viscosity, density, and viscosity. Reynolds

number is equal to the density multiplied by the velocity multiplied by the characteristic length divided

by the viscosity of the fluid. Although it is more commonly known to be the ratio between the inertial

forces to viscous forces in a fluid.

When an object is moving through the atmosphere, gas molecules close to the object are

disturbed and forced to move around the object. This creates aerodynamic forces between the object

and the air. The magnitude of these aerodynamic forces is determined by the shape and speed of the

object, the mass of the gas going against the object, and most importantly the viscosity and

compressibility of the gas flowing past the object. To accurately model the effects, similarity parameters

are used. Similarity parameters are the ratios of these effects to the other forces which are present in

the problem.

The Aerodynamic forces depend, in a complex way, on the viscosity (“stickiness” if the gas) of

the gas. A Boundary Layer is created by air molecules sticking to the surface of objects moving through

the air. The boundary layer can be a very confusing and difficult concept to grasp fully. A boundary layer

changes the physical shape of the object moving in the air. This is because the flow of gas reacts to the

edge of the boundary as if it is the physical surface of the object. The boundary layer is also capable of

separating from the body, and creating a more effective shape that may not resemble the shape of the

object. The boundary layer can separate and change at any time in the duration of the flight of the

object. There are no certain points in which it changes or separates from the body. The Boundary Layer

is essential in determining the drag of an object.

The Reynolds Number is also the most important similarity parameter for viscosity. “The

Reynolds number expresses the ratio of inertial (resistant to change or motion) forces to viscous (heavy

and gluey) forces.” (NASA). The inertial forces are the characterization of density, 𝑟, velocity, 𝑉, and the

gradient of the velocity 𝑑𝑉/𝑑𝑥, multiplied. The viscous forces are the characterization of the dynamic

viscosity coefficient, 𝑚𝑢, multiplied by the second gradient of the velocity, 𝑑2𝑉/𝑑𝑥2. The Reynolds

Number is then,

𝑅𝑒 =𝑟 ∗ 𝑉 ∗

𝑑𝑉𝑑𝑥

𝑚𝑢 ∗𝑑2𝑉

𝑑𝑥2


Ishan Amin 664854

Prandtl-Meyer Angle: When an object is in motion, the gas molecules around it are deflected. However, if the speed of the

object in motion is less than the speed of sound the gas retains a constant density, and the flow would

be described as conserving momentum and energy. If the objects increase towards the speed of sound,

the gas around the object may experience compressibility effects. The density of the gas varies due to

the fact that it is being compressed by the object in motion. When an object is able to move faster than

the speed of sound, and there is a decrease in the flow area, causing a shock wave. If, however, the

flowing air increases abruptly, a centered expansion fan is created.

A shock wave decreases Mach number, experiences an increase in static pressure, and due to the fact

that a shock wave is irreversible, there is a loss in total pressure. An expansion fan is the complete

opposite of the shock wave. An expansion fan increases the Mach number, witnesses a decrease in

static pressure and the total pressure is constant.

To calculate the variables of an expansion fan, the Prandtl-Meyer function is utilized. “This function is

derived from the conversion of mass, momentum, and energy for slight (differential) deflection.”

(NASA).

Angular Momentum: In rotational mechanics, all objects rotate around a specific point (for a Frisbee this point is the axis of

rotation). The radius of the orbit that the object is moving around or spinning

The angular momentum is the product of the moment of inertia and the angular velocity.

𝐿 = 𝐼 ∗ 𝜔,

where “𝐼” represents the moment of Inertia, and the 𝜔 represents the angular velocity. Momentum is

the measure of the force that an object has when it moves in a straight line, similarly, angular

momentum is how much force an object has when it moves in a circle (or to relate to the case, spins).

Vector Quantity: The quantity that has both a magnitude and direction used to determine one point in a plane, relative to

another. For example, the ball rolled 7 meters to the right. The magnitude, in this case, is 7 meters, and

the direction is right.

An evaluation on the physics of the flying disk

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