Relationship between Angle of Attack and Distance

Joseph Lin

Figure 3

The physics behind the Frisbee is surprisingly complex compared to its seemingly simplistic design. The behavior of the Frisbee once it is thrown is due to it's design that incorporates the essential physics concepts applied to airplanes. When a Frisbee is thrown, there are essentially three forces that act on it: weight, lift, and drag. Weight is the gravitational force that the Earth exerts on the Frisbee. For a Frisbee to sustain its vertical position in the air for a significant amount of time, a force must be present that counteracts the weight. This force is called the lift.

When you throw the Frisbee as flat and horizontal as you could, does it generally fly relatively straight and pretty far?

What if you throw the Frisbee tilted up a little, does it fly pretty high and far but not as straight, going off to the side near the end of its flight?

How about if you throw the Frisbee tilted down a little, does it fly not nearly as far—and does it also go off to the side?

Of the four launch angles tested in this activity, the horizontal launches probably resulted in the overall "best" Frisbee throws in terms of distance and straightness. However at an angle slightly tilted, there was a vast change in distance.

The question is WHY?

Newton's Laws

Many appeal to a model based on Newton's laws and assert that the main lift comes as a result of the ANGLE OF ATTACK. Part of the Newton's law model of part of the lift force involves the layer of air on the top of the wing with a resulting downward draft of air behind the wing. If the wing gives the air a downward force, then by Newton's third law, the wing experiences a force in the opposite direction - a lift.

Figure 1

Bernoulli's Principle

However, another side is calling on Bernoulli's Principle. Throughout the entire shape of the Frisbee, it is designed to utilize pressure differences within the air while in flight. When a Frisbee is thrown, due to the CURVATURE that is on the rim of the disc, it induces a HIGH pressure underneath the Frisbee while causing a LOW pressure above the disc thus enabling the Frisbee to fly effortlessly throughout the air.

This is essentially the exact principles that airplane wings are engineered after. Referring to Figure 1, the purple shape represents an airplane wing from the perspective of looking towards the body of the plane. As depicted, at different ANGLES OF ATTACK even slightly, a different and change in air flow occurs. This enables the plane to fly faster and further or land by slowing down and increasing the SEPARATION POINT by increasing the DRAG and air resistance.

Figure 2

Equations

Lift = W˔= mgcos(α) = ½ pv^2 A Cl

Drag = W|| = mgsin(α) = ½ pv^2 A Cd

m = mass (0.215 kg)

The lighter discs maximize duration of flight, and the heavier discs will maximize distance thrown and minimize the effects of wind and stray currents. It is also apparent from the equations of motion that a greater moment of inertia I would also increase stability.

g = gravity (-9.8 m/s^2)

α = angle of attack (relatively 0 degrees, <45 degrees, >45 degrees, relatively 0 degrees upside down)

p=Air Density (1.23 kg/m^3)

pv^2 = dynamic pressure

A = area

v = relative velocity (14 m/s)

Cd is the drag coefficient which according to research (Potts and Crowther) is

3.78 x 10^5, equivalent to speed of 20 m/s, for angles of attack from -10 degrees to 30 degrees

Before the trials, Simplicity was necessary due to time restraints and lack of experience. So factors such as spin(negligible) and the roll and pitch behaviors of the Frisbee in flight are not accounted for or calculated.

Also, due to the nature of video analysis, while useful for overall characterization of the experiment, by forward-modelling a trajectory with assumed parameters, the experiment would be subjected to significant errors in the determination of some point properties.

Trials

Trial 1: Frisbee thrown horizontally

Plot of height vs. distance for a Frisbee with initial velocity 14 m/s and angle of attack that is relatively 0 degrees

When the Frisbee is thrown relatively horizontal, it has a good amount of lift and consequently should fly relatively far. This shortness of distance can be explained by the Bernoulli's Principle of pressure. Due to the angle of attack that the Frisbee was thrown at, the levelness of the disc did not cause a major pressure change and so there was not enough LIFT enforced on the Frisbee.

Trial 2: Frisbee thrown at angle <45 degrees

Plot of height vs distance for a Frisbee with initial velocity 14 m/s and angle of attack that is less than 45 degrees

This time when the Frisbee is thrown at an angle slightly altered positively from 0 degrees, there is a considerably dramatic change in distance traveled done by the disc. The reasoning is due to the angle of attack relationship with the shape of the Frisbee. In this case, different from the previous trial, the angle of attack was angled in a manner that causes an ideal amount of LIFT for the disc to travel far as well as high. The angle of attack as depicted in Figure 2, causes a difference in air pressure around and underneath the Frisbee which enables maximum flight and height.

Trial 3: Frisbee thrown at angle >45 degrees

Plot of height vs. distance for a Frisbee with initial velocity of 14 m/s and angle of attack greater than 45 degrees

When an even larger launch angle is used, the Frisbee has more lift. You may have noticed, however, that although the Frisbee thrown upward flew relatively high, it probably stalled out rather abruptly near the end of its flight. This may have caused it to land gently and/or quickly go off to the side. In this trial, the Frisbee took the principle displayed in Trial 2 to the extreme by increasing the angle of attack dramatically, this forces a lift force that is too strong and causes the Frisbee to lose control and lose it's pressure difference that was enabling it to fly far. Thus the Frisbee drops.

Trial 4: Frisbee thrown upside down horizontally

To fly well, the Frisbee needs enough lift, and not too much drag. When the Frisbee is thrown tilted downward, it does not have much lift and so it quickly falls to the ground. This is also due to the shape of the Frisbee and how it incorporates pressure and velocity. Due to the lift force not being strong enough when the Frisbee is upside down, this induces an inverse parabolic arch that is much smaller than a Frisbee thrown right side up as depicted by Figure 2 previously.

Conclusion

In actuality, something as arbitrary as Frisbees is actually prevalent and a major part of our daily lives. It is the ingenious physics behind the flight of the disc that can be applied to multiple situations such as helicopters and airplanes. These modes of transportation has become an indispensable part of our lives. By mimicking and modeling, the Frisbee has shown what is possible in a small scale as well as grand.

Bibliography

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