Dynamics: Mass, Weight, and Inertia

Dynamics (2.2.3)

• Dynamics studies forces in motion and energy use.
• Energy is a fundamental property of the universe used to effect change; force is created when motion of a mass changes

Mass and Weight

• Earth's large mass attracts everything on its surface.
• Newton's Laws state F = ma (Force = mass x acceleration).
• Weight formula: W = mg (Weight = mass x acceleration due to gravity).
• Acceleration due to gravity (g) is 9.8 m/s² on Earth's surface.
• A person with a mass of 70 kg weighs approximately 700 N on Earth: F = 70 kg x 9.8 m/s² ≈ 700 N.
• Weighing scales technically should display newtons or kilograms-force (kg-f), with 70 kg-f equivalent to 70 kg on Earth.
• While mass remains constant, weight varies with gravitational force.
• On the moon, the same 70 kg mass weighs ≈ 114 N due to lower gravity (1.6 m/s²).
• On Jupiter, the same mass would weigh ≈ 1735 N due to higher gravity (2.5 times Earth's).

Inertia

• Inertia is a mass's property to resist changes in its state of motion.
• Newton's First Law states a body remains at rest or in uniform motion unless acted upon by an external net force.

Work

• Work is done when a force overcomes inertia and sets an object in motion.
• Zero work is done if there is no displacement.
• Work is calculated as: W = F x s (Force x distance).
• The SI unit for work is the joule (J), equivalent to 1 newton-metre (Nm).
• Example: Moving an object 10 m with a force of 100 N requires 1000 joules of work.
• Imperial unit of work is the foot-pound (the effort of raising 1 pound of mass by 1 foot).

Power

• Power is the rate of doing work, considering the time taken.
• Climbing stairs involves the same work whether walking or running, but running uses more power due to the faster rate.
• Power is calculated as: P = W/t (Work / time).
• The SI unit of power is the watt, equivalent to 1 joule of work done in 1 second.
• In the imperial system, power is measured in foot-pounds per second.
• 1 horsepower equals 550 ft-lb/s or 746 watts.
• Power can also be expressed as Power = Force x Velocity.

Energy

• Energy provides the capacity for work.
• The SI unit of energy is the joule.
• 1 joule of energy can perform 1 joule of work, assuming no energy losses such as friction.
• Energy can neither be created nor destroyed, only transformed from one form to another.

Potential Energy

• Potential energy is stored energy due to an object's position, condition, or chemical nature.
• An object can have potential energy even when not doing work, such as a mass held above the ground.
• Potential Energy formula: PE = mgh (mass x gravity x height).
• A drum of gasoline, a stick of explosive and a chocolate bar all contain potential energy because of their chemical composition.

Kinetic Energy

• Kinetic energy is the energy of a body due to its motion.
• Potential energy converts to kinetic energy as a held body falls.
• Kinetic Energy formula: KE = (1/2)mv² (½ x mass x velocity squared), measured in joules.

Total Energy

• Total energy in a system remains constant.
• Potential energy transforms into kinetic energy and vice versa.
• A falling mass has maximum potential energy at its highest point (PE = mgh) and zero kinetic energy.
• As it falls, potential energy converts to kinetic energy.
• Halfway through the fall, potential energy equals kinetic energy.
• At impact, kinetic energy is at its maximum, and potential energy is zero.

Heat

• Heat is a form of energy directly related to work.
• Other forms of energy can be transformed into heat.
• Heat is a consequence of friction, which is usually unwanted.
• The unit of heat in the International System of Units (SI) is the joule (J).

Efficiency

• Efficiency is the ratio of work output to work or energy input in any machinery.
• Efficiency is calculated as: (Work(out) / Work(in)) x 100%.
• Friction determines the efficiency of a machine, creating heat, sound, and light.
• Reducing friction can be done through lubrication or streamlining.

Friction

• Friction is the resistance when objects roll or slide in contact with other objects.
• Minimization of friction is sought with lubricant in most industrial applications.
• Friction between shoes and the ground is necessary to walk and run.
• Friction between tyres and brake rotors helps slow down a vehicle.

Coefficient of Friction

• Coefficient of Friction (COF) is represented by the Greek symbol µ.
• The COF is a dimensionless number representing the ratio of friction force to the force pressing two surfaces together and depends on materials used.
• The value of COF range from near 0 to greater than 1.
• The COF between surfaces of similar metals is higher than the COF between dissimilar metals.
• Lubrication reduces friction.
• Starting or static friction is the strongest type (overcoming initial resistance)
• Sliding friction is resistance during steady motion.
• Rolling friction is the force that resists the motion of a body rolling on a surface and is the weakest of the three types of friction (caused by bearing friction and tyre sidewall flex).
• Static friction, not rolling friction, is the friction between rolling tyres and bitumen.
• The amount of sliding friction is calculated as: F = µN, where F is the applied force, µ is the coefficient of friction, and N is the normal force.

Aircraft Landing Friction (Example)

• When pulling a box, it's easier at an upward angle compared to pushing it down.
• After an aircraft touches down, wings still support some weight reducing friction, and braking is inefficient between the wheels.
• To calculate sliding friction, use F = µN, where N = W - L (Weight - Lift).
• F = µ(W – L): Greater lift results in lower friction.
• Airflow spoilers reduce lift, allowing the pilot to begin braking earlier.

Rolling Resistance

• Coefficients of rolling resistance are very small.
• For example, rubber tyres on concrete has a coefficient of 0.02 and roller bearings have a coefficient of 0.001.
• Rolling one surface over another creates less friction than sliding one surface over another.

Momentum

• Momentum = the product of inertia and motion.
• Two types of momentum: linear and angular.
• Linear momentum: M = mv (mass x velocity). Tendency of a moving body to continue in motion along a straight line.
• Momentum is conserved: m₁v₁ + m₂v₂ = (m₁ + m₂) × V.

Angular Momentum

• Tendency of a rotating body to continue spinning about an axis.
• Formula: L = Iw, where L = Angular momentum, W = the rpm (or angular velocity) and I = the moment of inertia.
• Moment of Inertia (I) is the rotational inertia of a body, i.e., the opposition of the body to any alteration of its speed of rotation by an applied external force. It is calculated by multiplying the mass of each particle of matter in the body by the square of its distance from the axis. It corresponds to mass (m) in the formula for linear momentum.

Conservation of Momentum

• A spinning skater can change rpm by moving their arms (conservation of angular momentum).
• Angular momentum is increased or decreased by changing the distance of the skater’s arms from their body which raises or lowers resistance to spin.

Impulse

• Impulse: A force applied to a moving body changes its momentum.
• A spacecraft's burn is an example of impulse.
• Formula: F = (mv - mu) / t
  • Ft = mv - mu
  • t = time in seconds
  • mv = final momentum
  • mu = initial momentum
  • mv - mu = change in momentum
• Impulse measures change in momentum in Newton Seconds (Ns) or Kilogram meters per second (Kg.m/s).
• Impulse = Change in momentum = Ft.

Crumple Zones

• A large force multiplied by a small time, or a small force multiplied by a large time, can achieve a particular change in momentum.
• Reducing impact force of a rapid deceleration by spreading the impact over a longer time period
• Passengers in the car have the same change in momentum during a collision but with smaller forces (over a longer period of time) acting on their bodies.

Gyroscopes

• Gyroscope: Any rotating mass.
• A rotor mounted on gimbals allows a supporting platform or case to turn in one or more planes around the rotor without changing the rotor's plane of rotation.
• Two fundamental characteristics: gyroscopic inertia (rigidity in space) and precession.

Gyroscopic Rigidity

• Gyroscopic rigidity is the natural property of any rotating mass to resist changes to its plane of rotation unless an external force causes a change.
• Basis for the Artificial Horizon or Attitude Indicator.

Precession

• Precession is the change of the plane of rotation caused by an external force.
• When force is applied to the rotating mass, the plane of rotation deflects 90° in the direction of rotation
• Pushing the nose of this aircraft down causes the prop to swing the whole airframe left.