Topic 2: Mechanics
2.1 Motion
Vectors

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

Velocity (v 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 (v 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.
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 freefall (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."
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 k 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 μd is 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 m is the body's mass, a 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 actionreaction 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".

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
Freebody 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.
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 ForceDistance 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 Forceextension 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 forcetime 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
"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 = crosssectional area; l = cylinder's length; p = 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.