Aeroscience Lesson

For information purposes only! Not required to memorise

The International System of Units (or SI) is used for the measurement of physical quantities.

* Temperature can be expressed as degrees Celsius or degrees Kelvin. Zero degrees Celsius is equal to 273 K. One degree change in Celsius is equal to 1 degree change in Kelvin. Zero degrees K (−273°C) is the temperature at which all molecular movement stops.

Non-Standard Units in Aviation:

Due to the American influence on aviation, some units from the imperial system are used for the measurement of physical quantities.

The English physicist Sir Isaac Newton proposed three laws of motion in his work Principia Mathematica in 1687. The fundamental concepts in this work have formed the basis for all mechanical science ever since. Newton’s three laws of motion are as follows:

Newton’s first law: Every object will continue in a state of rest or uniform motion in a straight line, unless an external force acts upon it.

Newton’s second law: The external force acting on a body is proportional to the product of its mass and the acceleration produced by the force.

Newton’s third law: For every action, there is an equal and opposite reaction.

There is a natural tendency for things to continue doing what they are already doing. An object that is at rest will remain at rest, while an object in motion tends to continue at the same speed and in the same direction, until an outside force is applied. It can be stated that objects at rest or in uniform motion are in equilibrium (constant speed and constant direction) and possess the property known as inertia.

Inertia is a quality of an object.

Inertia is very closely associated with mass, which is a measure of the quantity of matter in an object. The inertia of an object is a measure of how difficult it is to get it moving or change the current state. The greater the mass of an object, the greater the force required to bring it to rest and, therefore, the greater the object’s inertia (i.e., its tendency to continue).

Mass is the amount of matter in an object.

A force is a push or pull. Force is identified by what it does.

While discussing mass and inertia, we stated that objects tend to continue doing what they are already doing until disturbed by an outside force. A force is required to overcome inertia.

Acceleration is the rate at which the push or pull can move an object out of a state of rest or uniform motion.

force (f) = mass (m) × acceleration (a)

All bodies have mass and will, therefore, experience the force of gravity acting on them. The rate at which a body gathers momentum or accelerates depends on its mass and the magnitude of the force acting on the body.

• The greater the mass of an object, the lesser the effect of any given force acting on the object; for example, compare trying to push a car with trying to push a bus with a given force—the car is far more likely to move!
• The greater the force, the greater its effect on any given mass.

Objects at the same distance from the centre of the earth may have different masses, but they will have corresponding different forces due to gravity acting upon them. Providing that the force of gravity is the only force that acts upon the objects, they will both accelerate at the same rate.

Gravity is the attraction between two objects.

In this formula, G is the gravitational constant: G = 6.67 × 10^-11 N (m/kg)²

If you take an object to 20,000 ft above the surface of the earth, the object’s mass will be the same, but its weight will be less. In other words, regardless of the object’s position, the quantity of matter in it (the object’s mass) remains constant. However, the further away an object is from the centre of the earth, the smaller the force of gravity acting upon the object; thus, its weight will vary. As per the formula above, if you double the distance between two objects, then the force of weight is reduced by a factor of four. If you take an object to the moon, the object’s weight will be reduced by 5/6.

The acceleration due to gravity is represented by the letter g. Be sure not to confuse G and g. For all masses on or near the earth’s surface, g is constant at 9.81 m/s².

Momentum is a quantity of motion and is the product of an object’s mass and velocity.

momentum (p) = mass (m) × velocity (v)

Mass is a measure of the amount of matter in an object. Where inertia is about the resistance of the mass to move, momentum is about the energy the mass has. Therefore, the velocity of the mass (speed and direction) must also be considered. If a mass is multiplied by velocity, the resultant quantity is known as momentum. Like velocity, momentum is a vector quantity, as it has direction and magnitude.

An object that has great momentum has a strong tendency to remain in motion and is, therefore, hard to stop. It is possible to have two objects of different mass with the same momentum, but as mass × velocity = momentum, the object with the smaller mass would have the greater velocity of the two. For example, a train has a great momentum due to its large mass, while a bullet has a great momentum due to its high velocity—they are both hard to stop!

Weight is a force.

To understand what weight is, it is necessary to define a force known as the force of gravity. There is a force of attraction between any two masses; this force is mutual (i.e., each mass attracts the other). On the earth, it is only necessary to consider the attraction between the earth itself and each body on or near the earth’s surface. In this form, the force of attraction can be termed weight. Thus, weight is the force produced when an object (or mass) is acted upon by gravity. The magnitude of this force depends on the mass of an object and its distance from the centre of the earth. The mass of an object will not change unless the matter is physically added or taken away from the object. However, as the distance from the centre of the earth increases, the force of gravity and, therefore, the weight will decrease slightly. As f = m × a, it follows that the acceleration here is the acceleration provided by gravity.

weight = mass (m) × gravity (g)

Centre of Gravity (Centre of Mass)

The centre of gravity is defined as the position of an object in which weight acts.

• When an object rotates, the centre of gravity is the centre of rotation.
• A suspended object so that it can move freely has its centre of gravity directly below the point of suspension.
• Objects balanced on a sharp point are placed directly beneath their centre of gravity.

Scalar Quantity

A scalar quantity has magnitude (size) but no direction (i.e., it is represented by a single point or number). We are only concerned with the amount of quantity or mass. Temperature, mass, and speed are examples of a scalar quantity.

Vector Quantity

A vector quantity has both magnitude and direction. A direction must be specified; otherwise, the vector quantity becomes a scalar quantity. Velocity, acceleration, and force are examples of a vector quantity.

A force is a vector quantity. It has magnitude and direction and can be represented by a straight line passing through the point at which the force is applied. The length of the line represents the magnitude of the force, and the direction of the line corresponds to the direction in which the force is acting.

Wind velocity is a vector quantity. As seen below, the line indicates the magnitude (wind strength) and direction (blowing from the west).

A force is a vector quantity (it has magnitude and direction) and can be represented by a straight line passing through the point at which the force is applied. The length of the line represents the magnitude of the force, and the direction of the line corresponds to that in which the force is acting.

As vector quantities, forces can be added or subtracted to form a resultant force. Forces can also be resolved (i.e., split up into two or more components by drawing the vectors to represent them).

Velocity and momentum are also vector quantities and can be represented by straight lines in the same way. Mass, on the other hand, is not a vector quantity, as mass has no direction.

Let’s look at some examples.

Two (or more) vectors acting in the same direction:

Two or more vectors acting in different directions can be combined into a single resultant by drawing them with the tail of one vector connected to the head of the other.

Two forces (vectors) acting from the same point can be drawn to find the resultant force.

The moment of a force about any point is the product of the force and the perpendicular distance (arm) from the point to the line of action of the force.

A moment about a point or pivot is a turning force that may act in a clockwise or anticlockwise direction. This rotational force is otherwise known as torque (τ).

moment or torque (τ) = force (f) × distance (d)

If a body is in equilibrium under the influence of several forces in the same plane, the sum of the clockwise moments about any point is equal to the sum of the anticlockwise moments about the same point; in other words, the total moment is zero.

When considering the forces acting on an object, two equal and opposite parallel forces are called a couple. Weight and lift form a couple; thrust and drag form a couple.

The moment of a couple is one of the forces multiplied by the distance between the two, i.e., by the arm of the couple. A couple may be defined as two equal and opposite parallel forces acting on a body that results in a torque or turning force.

When a force tries but does not succeed in altering the momentum of an object (i.e., it does not overcome inertia), it can be assumed that the applied force is being balanced by an equal and opposite force. Thus, one requirement for equilibrium is for all the forces to be equal and opposite. An aircraft can be considered to be in equilibrium at any time in which it remains at a constant speed while travelling in a constant direction (i.e., not turning and not accelerating or decelerating). This is also known as translational equilibrium, e.g., in straight and level flight when the turning moment of the lift and weight couple is equal to the turning moment of the thrust and drag couple.

Rotational Equilibrium

Although not strictly speaking equilibrium, an object is in rotational equilibrium when it has no angular acceleration and the sum of all the turning moments acting on it is equal to zero. An object in rotational equilibrium will be rotating at a constant rate or not rotating at all, e.g., a propeller that is either stationary or spinning at a constant RPM. All of the clockwise moments must be balanced by all the anticlockwise moments or, if torque exists in one direction, it must be balanced by a torque in the opposite direction.

Work

A force is said to do work on an object when it moves the object in the direction in which the force is acting. The amount of work done is measured by the product of the force and the distance moved in the direction of the force.

Work is only accomplished when an object has moved some distance against a resistive force, i.e., no matter how much pushing is done by a force, if it does not move the object, there is no work done. Similarly, if the object moves in a different direction to the line of action of the applied force, only the distance moved in the direction of the applied force can be used when calculating work. If the object is moved in the opposite direction or at right angles, once again, no work is done.

work (w) = force (f) × distance (d)

Power

Power is simply the rate of doing work.

For example, if a force of 10 pounds moves an object 2 feet in 5 seconds, then the power is 20 foot-pounds in 5 seconds or 4 ft-lb/sec. The unit of 1 ft-lb/sec is small for practical use and is usually converted into horsepower. Horsepower (hp) is defined as 550 ft-lb/sec or 33,000 ft-lb/min.

Energy

An object is said to have energy if it has the ability to do work. The amount of energy of an object is defined by the amount of work the object can do. The units for energy are the same as those used for work.

The amount of work an object can do should be the same as the amount of work it has had done on it. Due to friction, there is always loss of energy.

Energy can exist in many different forms: heat, sound, electrical, mechanical, pressure, etc.

Kinetic energy is the energy that results from motion. Every mass that is moving rapidly can do work in coming to rest and, therefore, has kinetic energy or the energy of motion.

Kinetic Energy = 1/2 Mass × Velocity²

Ek = 1/2 mv²

Potential energy is energy due to position. A weight that is high up can do work in descending and is said to possess potential energy or the energy of position.

Potential Energy = Mass × Gravity × Height

E_p = mgh

Conservation of Energy

Energy can never be destroyed. You can only change its state from one form to another. Consider a skydiver jumping out of a plane at 12,000 feet. At this altitude, the skydiver has a lot of potential energy (E_p). On departing the aircraft, the potential energy is converted into kinetic energy (E_k) as the skydiver accelerates towards the ground.

Speed

Speed is the rate of change of position or the distance travelled in unit time (i.e., metres/second, kilometres/hour, or nautical miles/hour).

Speed is a scalar quantity; thus, it does not have direction.

Velocity

Velocity is the rate of change of position in a given direction. This is the distance travelled in a given direction in unit time. Velocity differs from speed in that it has magnitude and direction; thus, it is the vector equivalent of speed. Velocity is a function of both speed and direction.

Acceleration

Acceleration is the rate of change of velocity of an object. Acceleration will occur whenever there is an increase in speed, a change in direction, or a change in both speed and direction.

An aircraft on the take-off roll is accelerating because it is increasing speed while maintaining a constant direction on the runway centreline.

Deceleration

Deceleration is negative acceleration, i.e., a decrease in speed.

An object that obeys Newton’s first law tends to remain in a state of uniform motion; thus, in order to make the object change direction, it is necessary to apply an outside force to it. While the force is applied, the object will describe a curve. But as soon as the force is removed, the object will continue in the direction it was headed when the force was removed. It will go off at a tangent.

It is important to realise at this point that an object moving on a curved path has acceleration towards the centre of its circle. Remember that velocity is a function of both speed and direction. So if the direction is altered, even though speed may remain constant, then the velocity is also altered, and with a change in velocity, there is acceleration.

Turning a moving object requires a force that is proportional to the mass and the acceleration required. The force acts towards the centre of the described circle. If an object is moving with uniform speed along a circular path, the object is constantly changing its direction and, therefore, will be undergoing acceleration. It was proved that there is acceleration on an object moving on a curved path, and this acceleration acts in the same direction as the force applied to turn the object (i.e., towards the centre).

A weight (e.g., a stone) being swung on the end of a piece of string describes a perfect circle, and the force that keeps it in the circle can be felt, and is supplied in the string. This force is known as the centripetal force, and the acceleration is known as centripetal acceleration.

The centripetal force is an inward-seeking force.

Although there is a centripetal force, the object does not approach the centre. So what holds the stone out then?

There must be another force acting on the stone equal to the centripetal force (because of Newton’s third law) but acting in the opposite direction. If the circle that the stone describes is thought of as an infinite number of straight lines, then at any given instant, the stone is travelling on a straight path. The stone has inertia and tends to continue straight.

However, the centripetal force is always present, which overcomes the stone’s inertia and forces it to accelerate on a curved path. If the centripetal force was suddenly removed, then the stone would, due to its inertia, continue in a straight line away from its curved path on a tangent to the circle.

Thus, there is a force that acts on the stone that is equal and opposite to the centripetal force; it is an inertia force. This force is only present when the centripetal force is applied. In reality, it is only a reaction. This inertia force is called the centrifugal force, and acts outwards in a turn.

The formula for the centrifugal reaction is the same as that for centripetal force. The two forces are equal in magnitude but opposite in direction.

Trigonometry is all about angles of a triangle and the length of their sides. You may remember this from your school days.

You will need to remember the following:

SOH CAH TOA

Let’s break it down:

SOH:

This refers to the angle is equivalent to the SINE of OPPOSITE divided by the HYPOTENUSE.

CAH:

This refers to the angle is equivalent to the COSINE of ADJACENT divided by the HYPOTENUSE.

TOA:

This refers to the angle is equivalent to the TANGENT of OPPOSITE divided by the ADJACENT.

Discuss the following questions in class now.

1. What are the units of measurement of force?
2. What are the key factors of Newton’s first law?
3. What are the key factors of Newton’s second law?
4. What is the key factor of Newton’s third law?
5. What is the formula for CPF?
6. Is CPF an inwards or outwards seeking force?
7. What is moment?
8. What is translational equilibrium?
9. Is an aircraft in a turn in equilibrium?
10. What is kinetic energy?
11. What is potential energy?
12. At what approximate height does the atmospheric pressure and density reduce to 75% compared to the sea-level value?
13. What is viscosity?
14. Using the formula for gravitational force, work out the force gravity is imparting upon you in newtons. Use your mass in kg. The radius of earth is approx. 6,378,100 m and its mass is approx. 5.98 × 10²⁴ kg.