Topic 2: Mechanics

2.1 Motion

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).

Projectile_motion.jpg

Fluid resistance effect on...

Projectile motion: 

  • Peak reduces in amplitude;

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

Parachutists: 

AirResistance.png
Parachute.png
  • 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."

2.2 Forces

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".

InclinedPlanesvg.png

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.

FreeBodyDiagram.png

Translational equilibrium

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

NetForce0.png
  • 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.

2.3 Work, Energy, and Power

 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.

 

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

 

totally inelastic

 

 

elastic and equal masses

 

 

 

elastic and unequal masses

  

TEste.gif
UnequalMasses.gif
inelastic.gif

"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.

ib-physics.net is a student initiative to provide free material to help international students prepare for the IB exams.

For any feedback or comments, contact a fellow IB alumni:

Maria Eduarda Lopes | mariaticilopes@gmail.com

Disclaimers: the IB organization does not endorse this website's content.

All images are under copyright that allows them to be shared.

(the source and copyright details may be downloaded by clicking on the images)