Vectors

  • Displacement (s): "change in position" - a "line" from the start until the end of the path.

  • Velocity (or u): displacement/time = ∆s/∆t.

  • Acceleration (a): change in velocity/time taken for the change = ∆v/∆t.

Scalars​

  • Distance (s): "length of path followed" - all twists and turns of the path included.

  • Speed (or u): distance/time = ∆s/∆t.

Speed or velocity can be...

  • Average: distance traveled over whole journey/time taken for whole journey.

  • ​​Instantaneous: short distance/small time interval.

  • Finding the gradient!

Motion

  • Uniform: constant velocity.

  • Uniformly accelerated: constant acceleration.

  • Not uniform: neither velocity or acceleration is constant.

Graphical representation

"SUVAT" Equations

Projectile motion

  • Decomposition of velocity into initial horizontal velocity (Vx) and initial vertical velocity (Vy).

  • Horizontal velocity remains constant during the projectile motion.

  • Vertical velocity can be calculated using the suvat equations, where the acceleration is acceleration of free-fall (g) and the displacement is height (h).

Fluid resistance effect on...

Projectile motion: 

  • Peak reduces in amplitude;

  • Peak shifts to the left (horizontal velocity reduces).

Parachutists: 

  • Terminal velocity: "the eventual constant velocity reached by a projectile (or a parachutist) as a result of an air resistance force that increases with velocity."

Inclined plane:

  • Weight is decomposed into a component horizontal to the plane and a component vertical to the plane.​​

  • Vertical: N= mg cos θ

  • Horizontal: ∑F = mg sin θ - Fd

Free-body diagrams

  • Illustration of all forces acting only on a body as vectors (Remember how to represent vectors).

  • All forces must be clearly labeled (e.g. Weight force/mg or Normal reaction force/R)

  • All forces must start at the center of the body.

Translational equilibrium

  • The body must be at rest or constant velocity, i.e. net force = 0 (circular motion not!)

  • Using tension: horizontal and vertical equilibrium.

    • T1 sin θ1 = T2 sin θ2

    • T = T1 cos θ1 + T2 cos θ2

Elevator issue

The reaction force is what a weighing scale measures. This is called the apparent weight.

 Principle of conservation of energy

"Energy is never created or destroyed, only transformed (e.g. into mass E = mc²), dissipated or transferred." Energy is measured in J (joules) - energy required to move 1 N through 1m.

  • ∆Esystem + ∆Esurroundings = 0

  • The energy of system changes as a result of interactions with the surroundings.

Work done (W) by a force

"The work done by a force is: force x distance moved in direction of the force"

  • W = Fs cos θ

    • The work done by a centripetal force is equal to zero, since the force is always at right angles to movement.

  • Graph: Work is also the area under a Force-Distance graph.

 

Energy (When work is done, energy is transferred)

  • Kinetic energy (Ek): energy related to motion - Ek = 1/2mv^2

    • Fractional change is the change of Ek divided ​by the original Ek.

    • Raised with constant speed - no net work done.

  • Potential energy (Ep): energy stored in a position.

    • Gravitational potential energy: energy related to height - Ep = mg ∆h                     

      • Independent of path followed ​- only ∆h matters ∆h.

    • Elastic potential energy: energy stored in a spring - Ep = 1/2kx^2

      • In a Force-extension graph, the area is the work done, and the gradient is k.

  • Other energies: Electric, Magnetic, Chemical, Nuclear, Thermal, Vibration, Light...

  • Dissipation: Energy transformed into thermal energy (internal energy of a body), sound.

 

Power (P)

  • "Power is the rate of energy transfer."  P = ∆W/∆t = ∆pv/∆t = Fv

  • Measured in W (watts).

Efficiency

  • Energy transferred = useful energy + wasted enery (never say lost energy!)

  • Efficiency = useful energy out/total energy in = useful power out/total power.

  • Efficiency is always smaller than 100% - frictional forces.

 

"Real life cases"

  • Recoil of a gun: Explosive "collision". Initial momentum = final momentum = 0.

    • Gun - higher mass, less speed; Bullet - less mass, higher speed.​

  • Water hoses​: A = cross-sectional area; l = cylinder's length; = H20 density.

    • Mass of water loss per second: pAl.

  • Rockets​: Explosive "collision"Mass thrown in one direction, rocket travels in the other

    • As total mass decreases, rate of increase of speed (acceleration) increases.​

  • Airbags: Increases person's head impact time - rate of transfer of momentum decreases (impulse remains the same) - Average force reduces.

Topic 2: Mechanics

2.1 Motion

Object as point particles

Object should be treated as point particles, i.e. dimensionless (as small as a point on a paper), unless otherwise stated.

Forces

  • Weight (W): W = mg, always directed downward; depends on location (e.g. Moon).

  • Tension (T)*: due to stretched strings, depends on the force exerted on the string.

  • Elastic (spring): F = kx (Hooke's law), where is the spring constant (in Nm^-1).

  • Normal reaction force (N or R)*: perpendicular to the surface of the body exerting the force.

  • Drag forces*: air resistance, fluid resistance - against motion.

Result of electromagnetic interactions between molecules.

Drag forces

  • Air resistance: normally proportional to the speed.

  • Friction: caused by asperities in the surfaces; not affected by area or speed.

    • Dynamic (when moving): Fd = μdR, where μis the coefficient of dynamic friction (a dimensionless scalar value).

    • Static (when not moving): Fs ≤ μsR , where μs is the coefficient of static friction, given that μs > μd and Fs is equal to the "pull", unless the pull is greater than μsR, in which case the object moves. 

 

Newton's laws of motion

  • First law (Principle of inertia): "An object continues to remain stationary or to move at a constant velocity unless an external force acts on it"

    • Consequence e.g.: Person in a car accelerating feels "thrown backwards​", because the body would naturally maintain its state of motion.

  • Second law: "F = ma" (simple form), where is the body's mass, its acceleration (normally measured in ms^-2) and F the force acting on it (measured in N - newtons).

    • Force and acceleration have the same direction, since they are both vectors.

    • Net force = Resultant force =  The sum of all forces =∑F

      • When the speed is constant the resultant force is equal to zero. 

  • Third law: "Every action has an equal and opposite reaction. The action-reaction pair must be of the same type". Hence, Fab = -Fba (Negative sign when against motion!)

    • E.g. Gravitational force: "Pull of Earth on man" and "Pull of man on Earth".

2.2 Forces

2.3 Work, Energy, and Power

2.4 Momentum and Impulse

  Basic concepts

  • Linear momentum (p): mass x velocity - "quantity of motion".

  • Impulse (I): change in momentum.

    • Derivation from Newton's Second law​ (assuming constant mass):

               ∑F = ma = m∆v/∆t = ∆p/∆t

  • Impulse = Area under force-time graph.

  • Units: kg m s^-1 or Ns

Principle conservation of linear momentum

"Momentum is always constant, if the net force on the system is zero (closed system)"

  • Kinetic energy may or may not be conserved in a collision. 

Collisions type

  • Elastic: Kinetic energy is totally conserved.

  • Inelastic: Kinetic energy is not conserved.

  • Totally inelastic (or plastic): Maximum kinetic energy lost - Bodies stick together.

  • Explosive: m1v1 = -m2v2  

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