Topic 2: Mechanics
2.1 Motion
Vectors
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Displacement (s): "change in position" - a "line" from the start until the end of the path.
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Velocity (v or u): displacement/time = ∆s/∆t.
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Acceleration (a): change in velocity/time taken for the change = ∆v/∆t.
Scalars
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Distance (s): "length of path followed" - all twists and turns of the path included.
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Speed (v or u): distance/time = ∆s/∆t.
Speed or velocity can be...
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Average: distance traveled over whole journey/time taken for whole journey.
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Instantaneous: short distance/small time interval.

Motion
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Uniform: constant velocity.
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Uniformly accelerated: constant acceleration.
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Not uniform: neither velocity or acceleration is constant.
Graphical representation
"SUVAT" Equations
Projectile motion
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Decomposition of velocity into initial horizontal velocity (Vx) and initial vertical velocity (Vy).
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Horizontal velocity remains constant during the projectile motion.
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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:
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Peak reduces in amplitude;
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Peak shifts to the left (horizontal velocity reduces).
Parachutists:
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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
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Weight (W): W = mg, always directed downward; depends on location (e.g. Moon).
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Tension (T)*: due to stretched strings, depends on the force exerted on the string.
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Elastic (spring): F = kx (Hooke's law), where k is the spring constant (in Nm^-1).
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Normal reaction force (N or R)*: perpendicular to the surface of the body exerting the force.
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Drag forces*: air resistance, fluid resistance - against motion.
* Result of electromagnetic interactions between molecules.
Drag forces
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Air resistance: normally proportional to the speed.
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Friction: caused by asperities in the surfaces; not affected by area or speed.
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Dynamic (when moving): Fd = μdR, where μd is the coefficient of dynamic friction (a dimensionless scalar value).
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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.
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Newton's laws of motion
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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"
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Consequence e.g.: Person in a car accelerating feels "thrown backwards", because the body would naturally maintain its state of motion.
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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).
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Force and acceleration have the same direction, since they are both vectors.
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Net force = Resultant force = The sum of all forces =∑F
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When the speed is constant the resultant force is equal to zero.
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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!)
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E.g. Gravitational force: "Pull of Earth on man" and "Pull of man on Earth".
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Inclined plane:
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Weight is decomposed into a component horizontal to the plane and a component vertical to the plane.
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Vertical: N= mg cos θ
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Horizontal: ∑F = mg sin θ - Fd
Free-body diagrams
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Illustration of all forces acting only on a body as vectors (Remember how to represent vectors).
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All forces must be clearly labeled (e.g. Weight force/mg or Normal reaction force/R)
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All forces must start at the center of the body.
Translational equilibrium
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The body must be at rest or constant velocity, i.e. net force = 0 (circular motion not!)
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Using tension: horizontal and vertical equilibrium.
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T1 sin θ1 = T2 sin θ2
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T = T1 cos θ1 + T2 cos θ2
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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.
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∆Esystem + ∆Esurroundings = 0
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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"
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W = Fs cos θ
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The work done by a centripetal force is equal to zero, since the force is always at right angles to movement.
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Graph: Work is also the area under a Force-Distance graph.
Energy (When work is done, energy is transferred)
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Kinetic energy (Ek): energy related to motion - Ek = 1/2mv^2
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Fractional change is the change of Ek divided by the original Ek.
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Raised with constant speed - no net work done.
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Potential energy (Ep): energy stored in a position.
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Gravitational potential energy: energy related to height - Ep = mg ∆h
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Independent of path followed - only ∆h matters ∆h.
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Elastic potential energy: energy stored in a spring - Ep = 1/2kx^2
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In a Force-extension graph, the area is the work done, and the gradient is k.
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Other energies: Electric, Magnetic, Chemical, Nuclear, Thermal, Vibration, Light...
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Dissipation: Energy transformed into thermal energy (internal energy of a body), sound.
Power (P)
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"Power is the rate of energy transfer." P = ∆W/∆t = ∆pv/∆t = Fv
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Measured in W (watts).
Efficiency
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Energy transferred = useful energy + wasted enery (never say lost energy!)
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Efficiency = useful energy out/total energy in = useful power out/total power.
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Efficiency is always smaller than 100% - frictional forces.
2.4 Momentum and Impulse
Basic concepts
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Linear momentum (p): mass x velocity - "quantity of motion".
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Impulse (I): change in momentum.
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Derivation from Newton's Second law (assuming constant mass):
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∑F = ma = m∆v/∆t = ∆p/∆t
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Impulse = Area under force-time graph.
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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)"
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Kinetic energy may or may not be conserved in a collision.
Collisions type
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Elastic: Kinetic energy is totally conserved.
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Inelastic: Kinetic energy is not conserved.
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Totally inelastic (or plastic): Maximum kinetic energy lost - Bodies stick together.
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Explosive: m1v1 = -m2v2
totally inelastic
elastic and equal masses
elastic and unequal masses
"Real life cases"
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Recoil of a gun: Explosive "collision". Initial momentum = final momentum = 0.
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Gun - higher mass, less speed; Bullet - less mass, higher speed.
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Water hoses: A = cross-sectional area; l = cylinder's length; p = H20 density.
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Mass of water loss per second: pAl.
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Rockets: Explosive "collision" - Mass thrown in one direction, rocket travels in the other
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As total mass decreases, rate of increase of speed (acceleration) increases.
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Airbags: Increases person's head impact time - rate of transfer of momentum decreases (impulse remains the same) - Average force reduces.