Math in Aviation - Essay Example

Math in Aviation "There are 5 aerodynamic forces that act on an airplane in flight ly lift (upward force), drag (Backward force due to air resistance), thrust (Forward force due to Engine power), gravity (Downward force due to weight) and centrifugal force. For airplanes to fly, the thrust (t) must be greater than the drag (d) and the lift (l) must be greater than the gravity (g) "(How Airplanes fly, Pg.1).. The conditions in mathematical language are t > d and l > g. The thrust is generated by the plane's engine. Drag: Air resists the movement of any object through it. This resistance is called drag. The drag set up by an airplane in steady level flight absorbs all the power development by the engine. Drag is the normal friction caused by the plane moving through air molecules. Drag is also a reaction to lift and this lift must be generated by the wings of the airplane in flight. There are some basic laws of motion which explains the aircraft fly. Newton's First Law of Motion: An object at rest will remain at rest or an object in motion will continue in motion at the same speed and in the same direction, until an outside force acts on it. For an aircraft to taxi or fly, a force must be applied to it. It would remain at rest without an outside force. Once the aircraft is moving, another force must act on it to bring it to a stop. It would continue in motion without an outside force. This willingness of an object to remain at rest or to continue in motion is referred to as inertia. Newton's Second Law of Motion: If a object moving with uniform speed is acted upon by an external force, acceleration will be directly proportional to the amount of force and inversely proportional to the mass of the object being moved. The motion will take place in the direction in which the force acts. Simply stated, this means that an object being pushed by 100 pounds of force will travel faster than it would if it were pushed by 50 pounds of force. A heavier object will accelerate more slowly than a lighter object when an equal force is applied. Newton's Third Law of Motion: For every action (force) there is an equal and opposite reaction (force). This law can be demonstrated with a balloon. If you inflate a balloon with air and release it without securing the neck, as the air is expelled the balloon moves in the opposite direction of the air rushing out of it. Bernoulli's Principle: When a fluid flowing through a tube reaches a constriction or narrowing of the tube, the speed of the fluid passing through the constriction is increased and its pressure is decreased. Lift: The lift force of a wing is derived from the independent action of its upper and lower surface. Aerodynamics is the study of the forces that let an aircraft fly. Motion is changing place or position. Air has no force or power other than pressure when it's motionless. When air is moving, its force becomes apparent. A moving object in motionless air has a force exerted on it as a result of its own motion. It makes no difference in the effect whether an object is moving in relation to the air or the air is moving in relation to the object. Newton's first and third Laws of Motion explain why deflecting the air downwards must result in a corresponding upward force. The air moving over the upper surface of the airplane is forced too travel farther, therefore velocity is increased. The increase in velocity caused a decrease in pressure. This causes lift in the upper surfaces. The air that passes beneath the airfoil has less distance to travel, this results in increase in air pressure on the lower surfaces and in decrease in air pressure on the upper surfaces. The total lift produced by the airfoil is equal to the difference downwards. The wing is said to have a relative angle of attack. The lift drops off at high angles of attack because the air instead of flowing smoothly over the upper surface breaks away from it and forms eddying currents. Gravity: Gravity is natural force acting upon the airplane. Gravity is the force that plays an important part in the performance of flight maneuver. The direction of gravity is always constant. It is exerted directly towards the center of the earth or downwards. Thrust: The forward force that the propeller develops is called thrust. It pulls or pushes the airplane forward through the air, overcoming the drag or the resultant of lift and gravity plus the drag. The thrust force developed by a propeller depends on the propeller design, the density of the air; and the speed at which it is turning. When the engine is idling by the thrust is negligible. "Centrifugal Force: Newton's first Law; a moving object will travel in a straight line indefinitely unless acted upon by an outside force. The law indicates that a force is required to compel any object to travel in a curved path" "( Hermine Smikle, Pg.1). Centrifugal force C.F. = mv2/r. Where m is the mass of the plane, V is the speed with the plane moves, R is the radius Mathematical concepts underlying the force in flight: "The lift and the drag of an airfoil section have a definite relation for any angle of attack. In addition the lift and drag depends on the angle of attack, contour of the wing, density of the air, area of the wing and square of the airspeed" ( Hermine Smikle, Pg.1).___ Lift l = CL. (V2/2).S Drag d = CD. (V2/2).A Where CL and CD are lift and drag coefficients which depend on the wing contour and the angle of attack. The term given in parenthesis is the dynamic pressure. S is the wing area. V is velocity or airspeed in ft per second. "Like lift, drag is proportional to dynamic pressure and the area on which it acts. The drag coefficient, analogous to the lift coefficient, is a measure of the amount of dynamic pressure gets converted into drag. Low drag coefficients are desirable because an aircraft's efficiency increases as drag decreases" (Theory of flight, Pg.1) The weight of an aircraft is a limiting factor in aircraft design. A heavy plane carrying heavy payloads, requires more lift than a light plane and more thrust to accelerate on the ground. On small aircraft, the location of weight is also important. A small plane must be appropriately"balanced for flight, for too much weight in the back or front can render the plane unstable. Weight can be calculated using a form of Newton's second law: W = mg Where W is weight, m is mass, and g is the acceleration due to gravity on Earth. Thrust itself is a force than can best be described by Newton's second law. The basic form of this law is: F = ma which states that force (F) is equal to mass (m) times acceleration (a). Acceleration is the rate of change of velocity over time. Thrust (T) is produced therefore by accelerating a mass of air. The maximum thrust is limited by the highest power output of the engine. Total thrust t + t2 = Drag d1 + left gravity resultant d2. When the airplane is flying at a constant rate of speed in level flight, the thrust pulling it forward must be exactly equal to the drag holding it back. References: 1. How Air Planes Fly. http://www.rc-airplane-world.com/how-airplanes-fly.html 2008. Page 1. 2. Hermine Smikle. Aerodynamics: The Mathematical Implications. 3. Theory of Flight. 1988. Page 1. Read More