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Basic Mechanics Kinematics

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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:                                   Fnett = p = pf – pi = mv – mu.
                                                   Fnett =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 = ½mvt2 = 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 = 2r/T.
     v2
                                                                       Fnet = ma
                                v1                                          = mv/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 = m2r/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/42.
       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‟}
                .: gM/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 = m42r/T2)
                        FG = FC,                                   FG = FC
               .: GMm/r2 = mv2/r,                           .: GMm/r2 = m42r/T2,
                      .: v2 = GM/r,                         .: GM/42 = r3/T2,

        So for a system with a given orbited object:       r3/T2 = GM/42 = 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= Fx,

                                    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 = 2r/T,
       Fnet= FW|| + FD + FF.                            aC= v2/r= 2r/T2
                                                        FC= maC= mv2/r= m2r/T2.
Momentum and Impulse
     p = mv.                                     Gravitation
     Fnett= p= pf – pi= mv – mu.                      g = GM/r2,
     Fnett =Area under F-t graph                       FG= GMm/r2,
                                                        FG = FC
Conservation of momentum                                v2 = GM/r,
      pA = – pB.                                      GM/42 = 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)