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Basic Mechanics (Kinematics) Kinematic Quantities Distance: x: length along a path [scalar]. Displacement: x: length in a single direction [vector]. Speed: v: rate of change of distance [scalar]. Velocity: v: rate of change of displacement [vector]. Instantaneous Velocity: vt: velocity at an instant in time. Initial Velocity: u: velocity at the start of a period of interest. Final Velocity: v: velocity at the end of a period of interest. Average Velocity: vav = (u + v)/2 Change in Velocity: v = v – u Acceleration: a: Rate of change of velocity. Equations of Motion (xuvat) x = ½(u + v)t. v = u + at. x = ut + ½at2 x = vt – ½at2. v2 = u2 + 2ax. Graphical Analysis Slope = derivative (Y/X), Area = integral (Y·X) Graph Type (Y-X) Slope Area displacement-time velocity - velocity-time acceleration displacement acceleration-time - velocity Vectors (trigonometry: adding perpendicular components) Pythagoras’ Theorem: H = √(O2 + A2). Triangular Relations: sin = O/H, H cos = A/H, O tan = O/A. A Frames of Reference (Newtonian Relativity) Inertial frame of reference: vref = 0. Relative velocity: vB:A = vB – vA (velocity of B relative to A). Newton’s Laws of Motion N1: A body will remain in a state of rest or constant velocity unless acted upon by an external (net) force (the sum of all forces acting upon that body). Law of Inertia: If Fnet = 0, then v = 0. N2: The acceleration of a body is directly proportional to the net force acting upon it and inversely proportional to its mass: The rate of change of momentum of a body is directly proportional to the net force acting upon it: Unbalanced Forces: Fnet = ma = p/t, p= mv. N3: The force exerted by one body onto another (action force) is equal and opposite to the force exerted by the second body back onto the first (reaction force). Action-Reaction Law: FAB = – FBA. Forces in two dimensions: Free-body diagrams: (normal force, weight, friction, driving force) To simplify diagrams of forces acting on a body, show all the different vector forces as arrows with lengths proportional to the magnitude of each force acting on a „particle‟ located at the centre-of-mass (xcm) with the mass of the body concentrated at the particle: xcm = (mixi)/mi = (m1x1+m2x2)/(m1+m2). FW = Weight = mg, g= gravitational field = g(Earth‟s Surface) ≈ 10 Nkg-1 =10 ms-2, FN = Normal Reaction Force = - FW┴, FD = Driving Force FF = Friction Force (opposes direction of motion) FN FF FD FW Inclined planes (parallel, ||, and perpendicular, ┴, components of weight) To find the net force on a body on an inclined plane, the vector forces can be resolved into two components: one perpendicular to the surface of the inclined plane (normal reaction force, FN, and perpendicular component of weight, FW┴), and one parallel to the surface of the inclined plane (driving force, FD, friction force, FF, and parallel component of weight, FW||): FN = - FW┴ = - mg·cos, FW|| = mg·sin, Fnet = FW|| + FD + FF, (at an angle of up the slope). FN FN + FW┴ = 0 FF FW|| + FD + FF = Fnet, Fnet FN FW|| N.B. FF is opposite to the other FF FD parallel forces and so is FW|| FW┴ negative; i.e., subtracted from FD FW┴ the other parallel forces! FW Collisions: Momentum and Impulse Momentum: p = mv. Impulse: Fnett = p = pf – pi = mv – mu. Fnett =Area under F-t graph Peak or average Force acting on an object increases as time interval decreases. Fnet p1 = p2 Fnet Area 1 = Area 2 t t Collisions: Conservation of Momentum Isolated System: A system of objects in which the only forces acting on the objects are the action-reaction forces between the objects in the system (i.e., no external forces acting on the system): Newton’s Laws of Motion (N1, N2 & N3) for Momentum: N1: If Fnet = 0, then v = 0. & N2: Fnet = p/t. .: If Fnet = 0, then p = 0. N3: FAB = – FBA. .: pA = – pB. Conservation of momentum: In an isolated system the total momentum of all objects in the system remains constant: pT(iso) = pTf – pTi = 0. pcm(iso) = 0, xcm=(mixi)/mi= (m1x1+m2x2)/(m1+m2). Collision Equation: pTf = pTi mAuA + mBuB = mAvA + mBvB. Energy, Work and Power: Energy Transfer and Transformation Conservation of Energy: Energy cannot be created or destroyed; it can only be transferred or transformed [scalar]: ET = W + EK + EGP + EEP (+Q), Q= heat, sound, light, fracture, etc... Work: Energy transferred or transformed from one object to another by the application of a force. In mechanical systems, the work done by a net force in moving an object in the direction (x·cos= x||) of that force is equal to the change in kinetic energy of that object: W = Fnetx·cos= max|| = EK. Kinetic Energy: Energy associated with the motion (v) of an object (m): EK = ½mvt2. EK = ½mvt2 = ½mv2 - ½mu2. Potential Energy: Energy available to do work due to the position or configuration of an object and its surroundings. Gravitational Potential Energy: Potential energy stored in an object (m) due to its position (h) within a gravitational field (gravitational field strength = g): The energy available to do work by the force of gravity (weight = FW = mg) acting across the distance (height = h) between the object and some reference point below it (usually the ground): EGP = mgh, (special case of W= max||, a=g, x||=h). Elastic Potential Energy: Potential energy stored in an object as a result of a reversible change of shape. The energy available to do work by the restoring force (FEP) of the object acting across the distance (x) of its compression: Hooke’s Law: FEP = –kx, EEP = ½kx2 = Area under FEP-x graph, x = compression. Power: The rate at which energy is transferred or transformed: P = W/t = Fv. Elastic & Inelastic Collisions Elastic Collisions: Conservation of Kinetic Energy; EKec = 0. Inelastic Collisions: Non-conservation of Kinetic Energy; EKic ≠ 0. Projectile Motion (Ignoring Air Resistance) Projectiles travel in parabolic path (no air resistance), because the only force acting on a projectile is gravitational force, FW = mg, providing a constant vertical acceleration, aV = g ≈ 10 ms-2, and no horizontal acceleration, aH = 0 ms-1. To analyse projectile motion, find the vertical and horizontal components separately. You may also need to calculate the upward and downward arcs (paths) separately. Initial Velocity at an angle: Vertical: uV = u·sin, Horizontal: uH = u·cos, Projectile Path: (x = ut + ½at2) Vertical: xV = uVt + ½aVt2, aV = g (no air resistance). Horizontal: xH = uHt, aH = 0 (no air resistance). Special Cases (Problem Solving Hints): Maximum Height: at xVmax, v = 0 .: xVmax= ½gt2 (no air resistance). Total time: Calculated from the vertical component (time to ground). Range: Total time used to calculate the total horizontal displacement. vtop = vH xVmax= ½gt2 u uV u=v uH xHmax uV = u·sin xV= uVt + ½gt2 aV= g ≈ 10 ms-2, vVtop= 0 uH = u·cos xH= uHt aH= 0 ms-2, vH = uH Circular Motion The period (T) of a repeated circular motion is the time taken for one revolution. Velocity: v = x/t = circumference/period: Uniform circular motion: v = 2r/T. v2 Fnet = ma v1 = mv/t v = m(v2-v1)/t -v1 .: Fnet and a are v2 a = v/t directed towards a = v/t the centre of the = (v2 -v1)/t circle of motion. Centripetal acceleration: centre-directed acceleration of an object moving in a circle: aC = v2/r = 2r/T2, {from a= v/t and v:v=u::x=vt:r}, Centripetal force is the centre-directed force acting on an object moving in a circle. In the case of a circular motion with constant speed, the centripetal force is equal to the net force: FC = maC = mv2/r = m2r/T2. Examples of Centripetal Force: Tension and Friction (horizontal and at an angle). Universal Gravitation Overview Gravitational field strength: g = GM/r2. Gravitational force: (Newton‟s Law of Universal Gravitation): FG = Gm1m2/r2. Orbital constant: r3/T2 = GM/42. Changing gravitational field: Energy change = area (Gravitational force–radius graph). Gravitational Field Strength: Gravitational field strength (g) is the force of gravity on an object of unit mass. g M, M = mass of orbited object g 1/r 2 r = radius from centre of orbited object {inverse square law: influence proportional to radiating „surface area‟} .: gM/r2, g = [Nkg-1], {F = ma, a = F/m [ms-2; Nkg-1]} M = [kg], r2 = [m2], .: proportionality constant = gr2/M = [Nm2kg−2], Gravitational Constant: G = 6.67 x10-11 Nm2kg−2. The gravitational field strength „g‟ at a distance „r‟ from a body of mass „M‟ is given by the formula: g = GM/r2. Newton‟s Law of Universal Gravitation: Combining Newton‟s 2nd Law with the formula for gravitational filed strength: N2: Fnet = ma, a = g = GM/r2. The force of gravity „FG‟ on an object of mass „m‟ at a distance „r‟ from another object of mass „M‟ is therefore given by: FG = GMm/r2. This equation is referred to as “Newton‟s Law of Universal Gravitation”. c/f Coulomb‟s Law (Electrostatic Force). Balancing Centripetal and Gravitational Forces: The gravitational force of an orbited object on a orbiting object provides the centripetal force of the orbiting object, .: Gravitational Force (FG = GMm/r2) = Centripetal Force (FC = mv2/r = m42r/T2) FG = FC, FG = FC .: GMm/r2 = mv2/r, .: GMm/r2 = m42r/T2, .: v2 = GM/r, .: GM/42 = r3/T2, So for a system with a given orbited object: r3/T2 = GM/42 = constant. Changing gravitational field: The kinetic energy transferred to/from a satellite can be determined from the area under a graph of the gravitational force on the satellite versus the radius of its orbit: FG = GMm/r2, FG .: FG 1/r2. W= Fx, W= area(F-x graph) r Motion in Two Dimensions: General Summary Equations of Motion (xuvat) Potential/Kinetic Energy and Work x = ½(u + v)t. W= Fnetx·cos= max|| = EK. v = u + at. EK= ½mvt2. x = ut + ½at2 EGP= mgh, W= max||, a=g, x||=h. x = vt – ½at2. FEP= –kx, v2 = u2 + 2ax. EEP= ½kx2 = Area under FEP-x graph, Graph Slope Area P = W/t = Fv. x-t v - EKec = 0. v-t a x EKic ≠ 0. a-t - v Projectile Motion Newton‟s Laws of Motion uV = u·sin, N1: Law of Inertia: uH = u·cos, If Fnet = 0, then v = 0. xV = uVt + ½aVt2, N2: Unbalanced Forces: xH = uHt, Fnet = ma = p/t, p= mv. aV= g ≈ 10 ms-2, N3: Action-Reaction Law: aH= 0 ms-2, FAB = – FBA. vVtop= 0 vH = uH Inclined Planes FN= - FW┴= - mg·cos, Circular Motion FW||= mg·sin, v = 2r/T, Fnet= FW|| + FD + FF. aC= v2/r= 2r/T2 FC= maC= mv2/r= m2r/T2. Momentum and Impulse p = mv. Gravitation Fnett= p= pf – pi= mv – mu. g = GM/r2, Fnett =Area under F-t graph FG= GMm/r2, FG = FC Conservation of momentum v2 = GM/r, pA = – pB. GM/42 = r3/T2 = constant pT(iso) = pTf – pTi = 0. W= area(F-x graph) mAuA + mBuB = mAvA + mBvB. Problems Types: Describing (relative or vector) motion (formulae, graphs, & words) Balancing vector forces (action-reaction forces, inclined planes) Collisions impulse-momentum (maximum/average forces) Conservation of momentum collisions (isolated systems) Energy transformation (work done to change in kinetic energy) Energy transformation (gravitational/elastic potential storage/release) Elastic inelastic collisions (non-/conservation of kinetic energy) Projectile motion (speed and angle of take-off for a distance travelled) Centripetal motion (mass, acceleration, velocity, radius, period, force) Gravitational field strength (inverse square law, gravitational constant) Balancing gravitational and centripetal forces of objects (velocity, radius) Describing orbital motion (period and radius for a given orbital system) Work done shifting an orbiting object (changing gravitational field)

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posted: | 3/25/2011 |

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