The Wright brothers flew the first manned aircraft over 100 years ago, but people have been fascinated by the idea of flight for thousands of years. Leonardo DaVinci sketched an early design for a helicopter in 1493:
We now have a very good understanding of what it takes to fly. How does an aircraft lift off of the ground? How does it fly once it's in the air? There is a lot of physics behind those two questions, and it's not as hard to understand as you might think. This tutorial will take you through the basic physics principles used by multirotor aircraft such as quadcopters and sUAVs to fly and maneuver through the air. Understanding those principles will make you a better operator overall and you can apply this knowledge to other flight systems as well. This tutorial is a basic introduction to the physics, and more information and practice problems are available by following links listed throughout the tutorial.
There are forces all around us - the force of gravity pulls us downward, wind pushes from the side, magnets pull on metal, and water pushes up a beachball to make it float. A force is any kind of push or pull - when a force is applied to an object, that object speeds up.
You might be thinking, “if forces make me speed up, then why isn’t gravity making me speed up?” The answer is that there is another force, the ground pushing you back up with an equal amount of force. The result is that you do not move.
Objects almost always have a set of forces that act on them at any given time. In order to determine how much an object will speed up or slow down and in what direction, you’ll have to determine each of the individual forces that are acting on an object and add them all together.
Adding two forces together isn’t as easy as adding two numbers together because forces are more than just a size. Forces also have a direction. To represent a force, we use a vector. A vector contains both size and direction, and can be drawn using an arrow. The arrow can point in any direction and its length represents the size of the force. Other things can be represented with vectors as well. When speed and direction are combined, it is called velocity, and velocity can be represented with a vector.
Using vectors, we can add the forces of two soccer players kicking a soccer ball at the exact same time, and predict which way soccer ball will go:
Where did the 5 pound answer come from? Hint: re-arrange the 3 and 4 pound vectors to make a triangle. Then, use the Pythagorean Theorem [1] to find the length of the result. If you know how to calculate angles with Tangent [2], you can find the exact direction of the resulting force as well!
Below are a few problems to help you practice learning how to use vectors to find resultant velocities.
Try creating your own scenarios like the ones in the problems above. Remember that vectors can be used to represent anything that has both size (magnitude) and direction, including velocities and forces. What other measurable properties can be represented by vectors?
When an object changes its speed, it’s called acceleration. The larger the change in speed, the larger the acceleration. We can also predict how much the soccer ball will accelerate during the kick if we know something about the soccer ball: its mass. Mass is the amount of matter in an object. A bowling ball has more matter in it than a soccer ball, therefore the bowling ball has more mass than the soccer ball:
If you apply the same kick to both the soccer ball and the bowling ball, the soccer ball will speed up much more than the bowling ball during the kick (don’t try kicking a bowling ball - you could break your foot!) When mass increases, acceleration decreases. We can represent this as:
For the same force, a large mass only accelerates a small amount, and a small mass will accelerate a large amount:
The following experiement will help you understand the relationship between force, mass, and acceleration. To do this experiment, you will need the following materials:
Effect of force on mass:
Observe that when mass is increased, the same force cannot accelerate (and therefore move) the blocks as far. Increasing the mass decreases the acceleration.
Effect of force on acceleration:
Observe that the block moves the longest distance when the ruler's incline is steepest. This is because the steepness of the ruler increases the acceleration of the ball, which in turn increases the force it can apply to the block.
Can you think of additional ways to test the relationship between force, mass and acceleration?
Educators - Additional resources are available on Khan Academy: One-dimensional motion [3], Two-dimensional motion [4] and Forces and Newton’s Laws of motion [5]. Also, search online for "F=ma practice problems [6]" to find additional worksheets, problems, and projects.
Forces change an object’s motion, but without them, an object will keep doing whatever it was doing. If the object is not moving, it will stay in place. If an object is moving, it will keep moving at the same speed in the same direction forever unless a new force changes or stops its motion. An object’s tendency to keep doing whatever it is doing is called inertia. An object’s mass determines how much inertia it has. In fact:
Once an object is moving, it takes some force to stop it or change its motion. The more velocity it has, the more force it takes to stop. In addition, the more mass an object has, the harder it is to stop. The combination of mass and velocity is called momentum:
Momentum is a measure of how much movement an object has, and knowing an object’s momentum can help you determine how much force it will take to stop or change the direction of a moving object. Changing the motion of an object requires a force to be applied for a certain amount of time. A force applied for an amount of time is called an impulse:
Imagine pushing a car in neutral. If you push with 10 pounds of force for 10 seconds, or push with 100 pounds of force for 1 second, the speed it will end up moving with will be the same. An impulse applied to an object gives it momentum. In fact, an impulse results in a change in momentum:
What momentum doesn’t help determine is how much energy is contained in the movement of an object. An object’s Kinetic Energy is determined by half of its mass times the square of its velocity:
Because the velocity is squared (times itself again), an object that is moving 100 miles per hours has 4 times as much kinetic energy as an object that is only moving 50 miles per hour. This is important to know because it is an object’s kinetic energy that describes things like how long it will take to stop and how much damage it will do in a collision.
Educators: Additional Lessons and Resources are available on Khan Academy here: Impacts and linear momentum [7], and practice problems can be found by searching online for "momentum and impulse practice problems [8]"
There are a lot of different things that fly, some of them in nature and others that are man-made. Each of them use the same set of forces to help them get off the ground and into the air.
The simplest flying man-made object is a balloon. By filling a very light balloon with something that is lighter than air, such as hot air, hydrogen, or helium, the balloon will float. Gravity pulls down on the balloon, but not very hard because the balloon is very lightweight. A helium balloon also experiences lift because of something called buoyancy.
The problem with a helium balloon is that there is very little that can be done to control where the balloon goes other than up. Imagine attaching a couple of fans to the balloon. A large enough balloon will balance out the additional weight of the fans, and the fans will provide thrust. Thrust is a force that pushes the balloon in a desired direction. When the balloon begins moving forward from the the thrust, the balloon will also begin to experience drag.
As silly as this drawing may look, it is exactly how airships work: Drag is the effect of air resistance on the balloon. When the balloon begins moving through the air, it has to push the air out of the way - and the air pushes back. You can feel air resistance if you move your hand quickly through the air. The faster an object moves through the air, the more air resistance it will experience. Air resistance, or drag, is always a force in the opposite direction of the object’s movement:
The amount of drag an object experiences depends on a number of factors:
The actual formula is:
Educators: for more information, The Physics Classroom has additional information on free-fall and air resistance here [9]. Search online for "air resistance practice problems [10]" for additional worksheets and problem sets.
All flying objects experience the same basic four forces: Weight, Lift, Thrust, and Drag. By controlling one or two of these forces, aircraft can fly and move through the air. Different types of aircraft use different engines and mechanisms to generate and change these forces, and each method has advantages and disadvantages. There are several different types of aircraft:
An airplane’s lift is created by its wings. Wings on an airplane have a special shape (similar to the wing of a bird) that generates lift when it moves forward through the air. The shape of an airplane wing is called an airfoil. The top of an airfoil is longer than the bottom. The longer top reduces the pressure of the air above the wing, and since the air below wing is at a higher pressure, it pushed the wing up, generating lift:
By tilting the wing or changing the shape of the wing, it can change the amount of lift it generates, allowing to plane to go up or down (or if one wing pushes up and one pushes down, to tilt or turn).
Because a plane cannot lift without moving forward, it is not possible for most airplanes to hover. Hovering means staying still in mid-air. Helicopters are able to hover because instead of moving the whole aircraft, they only move the wing. On a helicopter, the wing is called a rotor, and it is spun rapidly in order to generate the lift needed for the helicopter to fly.
All of that rotation is actually a problem for the helicopter. Without some additional help, the rotation of the helicopter’s rotor will cause the helicopter itself to begin rotating in the opposite direction (more on that later...)
Do you remember all of the forces acting on the balloon-fan aircraft?
In the balloon-fan aircraft illustration, you can see that the weight of the aircraft is being counteracted by the balloon's lift, and some of the thrust is being counteracted by the drag.
Just like in the soccer ball example, the forces in the balloon-fan aircraft are all added together to find the remaining resultant or net force acting on the aircraft.
Educators: for more information, The Physics Classroom has additional information on the addition of forces here [11]. Search online for "air resistance practice problems [12]" for additional worksheets and problem sets.
By varying the amount of lift produced on different parts of the aircraft, the aircraft can tilt or turn to change its motion. On an airplane, there are a set of flaps called control surfaces that alter the amount of lift and/or push on the air flowing past the airplane, causing it to tilt or turn.
The set of flaps at the rear edge of the wings on an airplane are called ailerons. They are used to tilt the airplane sideways by causing one wing to push up and the other wing to push down. This motion is called roll. Rolling allows the airplane to turn:
Multirotor aircraft like quadcopters don’t have control surfaces like airplanes so, but they are able to roll by speeding the motors up to generate more lift on one side and slowing down the motors to generate less lift on the other, causing a controlled tip that results in a roll:
When a multirotor aircraft rolls, it begins moving sideways, which is a type of lateral movement. Lateral movement refers to any movement that is parallel to the ground. Another type of lateral movement is caused with a multirotor pitches. Pitch refers to the amount of forward and backward tilt. On an airplane, control surfaces called elevators are used to push up or pull down on the tail of the airplane, causing it to tilt up or down. Airplanes use this movement to climb or dive:
Multirotors use pitch to move forward or backward:
Remember how forces are vectors? When the multirotor is rolled to the side, some of its thrust is now pulling it sideways, and less of it is pulling it upward:
Just like two forces can add together to become a single force in a new direction, a single force can be broken into multiple component forces - for instance, we can break down the thrust of a tilted multirotor to analyze both the part of the thrust that is still providing lift, and the part of the thrust that is pulling the multirotor to the side.
Notice that the part of the thrust that is lifting the multirotor is smaller? If the total amount of thrust is not increased, it will begin to fall because the thrust is no longer providing enough lift to counteract the pull of gravity:
How much additional thrust is necessary? To answer that question, we'll use some right-angle trigonometry. First, we have to know how much thrust is being put out by the quadcopter. If it is hovering, then the amount of thrust is equal to the weight of the quadcopter (remember: forces add up, and weight and thrust/lift cancel each other out if it's not accelerating upward or downward.) For this example, we will say that the thrust is 3 pounds.
The other thing we need to know is the new angle of the thrust. For this example, we will use a 30° tilt.
Draw out the triangle with the angle of 30° at the top and the vertical leg with a length of 3:
Next, we'll use the cosine function to determine the length of the side representing the thrust. The cosine function describes the relationship between an angle and the ratio of the side adjacent to the angle and the hypotenuse (the longest side of a right triangle). When we do the math, we get:
This means that an additional 0.46 pounds of thrust is required to keep the quadcopter at the same height.
We can use this same type of math to find out how large or small the sideways (horizontal) component of the thrust is. Just like the cosine function describes the relationship between the adjacent side and hypotenuse, the Tangent function describes the relationship between the adjacent and opposite sides of the triangle. If we find the length of the opposite side, we will know how large the sideways component of the thrust is:
Try calculating the additional thrust necessary for multirotors of different weights with tilts at diffferent angles. If your calculator has a cosine (abbreviated cos) button, make sure your calculator is in degree mode.
Try making paper airplanes with differnt types of control surfaces. Bend or fold the control surfaces in different ways to observe how your paper airplane pitches, rolls, and tilts.
Educators: for additional resources and practice problems, search online for "forces at an angle [13]"
When a quadcopter tilts so that it can direct its thrust and move laterally, it does so by creating more thrust on one side of the quadcopter than the other. This difference in thrust creates a torque. A torque is a force applied some distance away from a rotating center:
Torque is greatest when the force is applied further away from the center - this is why a long wrench helps you loosen a bolt more easily than a short wrench. To calculate torque, we use the formula:
Torques are what cause rotation. Rotation is when an object turns around its center. The faster something rotates, the more angular velocity it has. Angular velocity can be measured in degrees per second. Another common measurement is revolutions per minute, or RPM. If you remember that there are 360° in a circle, and if an object takes 2 seconds to make one complete rotation, it's angular velocity would be 180° per second.
Do you remember that when an object is moving quickly it has a lot of momentum? The momentum from earlier in the chapter applied to objects moving in a straight line. What if an object is sitting still but rotating like a top? This is another kind of momentum called angular momentum. Just like momentum, the more quickly an object is rotating, the larger its angular momentum is. Momentum and angular momentum are both conserved. That means that if one part of an object starts rotating, another part of it must rotate in the opposite direction unless a force prevents it from rotating in that opposite direction. Imagine a drill hanging from a string. If the drill is activated, the bit will spin rapidly. The drill’s body will conserve angular momentum by rotating in the opposite direction:
Helicopters experience this as well. By using a tail rotor, helicopters are able to prevent their bodies from spinning out of control:
Some larger helicopters (like the Boeing Chinook CH47 pictured below) use two rotors that spin in opposite directions so that the angular momentum of one rotor cancels out the angular momentum of the other:
Educators: For additional resources, visit Khan Academy here [14] or search online for "angular momentum practice problems [15]"
Airplanes are able to turn by using a control surface called a rudder. The rudder is used to push the tail to one side or the other, causing a rotation referred to as yaw. Yawing allows an aircraft to turn similar to how the front wheels of a car are used to make the car steer:
Multirotors are able to yaw by taking advantage of the angular momentum of the spinning motors and propellers. Remember that on a quadcopter, 2 of the motors spin in one direction and two of them spin in the other direction to balance out the angular momentum on a quadcopter. If you intentionally unbalance the angular momentum by speeding the two clockwise spinning motors up and slowing the two counter-clockwise spinning motors down, the body of the quadcopter begins turning counter-clockwise:
It is important to note that when a quadcopter is yawing, only two of the motors are providing most of the lift. This means that the quadcopter is less stable during a yawing maneuver and more susceptible to interference from wind or disruption from other maneuvers such as roll and pitch that might be happening at the same time.
Links
[1] https://www.mathsisfun.com/pythagoras.html
[2] https://www.mathsisfun.com/definitions/tangent-function-.html
[3] https://www.khanacademy.org/science/physics/one-dimensional-motion
[4] https://www.khanacademy.org/science/physics/two-dimensional-motion
[5] https://www.khanacademy.org/science/physics/forces-newtons-laws
[6] https://www.google.com/search?q=f%3Dma+practice+problems&gws_rd=ssl
[7] https://www.khanacademy.org/science/physics/linear-momentum
[8] https://www.google.com/search?q=momentum+and+impulse+practice+problems&gws_rd=ssl
[9] http://www.physicsclassroom.com/class/newtlaws/Lesson-3/Free-Fall-and-Air-Resistance
[10] https://www.google.com/search?q=air+resistance+practice+problems
[11] http://www.physicsclassroom.com/class/vectors/Lesson-3/Addition-of-Forces
[12] https://www.google.com/?gws_rd=ssl#q=net+forces+practice+problems
[13] https://www.google.com/search?q=forces+at+an+angle
[14] https://www.khanacademy.org/science/physics/torque-angular-momentum
[15] https://www.google.com/search?q=angular+momentum+practice+problems