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P1X*Dynamics & Relativity : Newton & Einstein

P1X*Dynamics & Relativity : Newton & Einstein. Part I - “I frame no hypotheses; for whatever is not deduced from the phenomena is to be called a hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy.”.

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P1X*Dynamics & Relativity : Newton & Einstein

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  1. P1X*Dynamics & Relativity:Newton & Einstein Part I -“I frame no hypotheses; for whatever is not deduced from the phenomena is to be called a hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy.” Dynamics Motion Forces Energy & Momentum Conservation Simple Harmonic Motion Circular Motion READ the textbook! section numbers in syllabus http://ppewww.ph.gla.ac.uk/~parkes/teaching/DynRel/DynRel.html Chris Parkes October 2004

  2. Motion x e.g dx • Position [m] • Velocity [ms-1] • Rate of change of position • Acceleration [ms-2] • Rate of change of velocity 0 t dt v 0 t a 0

  3. Equations of motion in 1D • Initially (t=0) at x0 • Initial velocity u, • acceleration a, s=ut+1/2 at2, where s is displacement from initial position v=u+at Differentiate w.r.t. time: v2=u2+2 as

  4. 2D motion: vector quantities Scalar: 1 number Vector: magnitude & direction, >1 number • Position is a vector • r, (x,y) or (r,  ) • Cartesian or cylindrical polar co-ordinates • For 3D would specify z also • Right angle triangle x=r cos , y=r sin  r2=x2+y2, tan  = y/x Y r y  x 0 X

  5. vector addition y b • c=a+b cx= ax +bx cy= ay +by a c can use unit vectors i,j i vector length 1 in x direction j vector length 1 in y direction x scalar product a  finding the angle between two vectors b a,b, lengths of a,b Result is a scalar

  6. Vector product e.g. Find a vector perpendicular to two vectors c Right-handed Co-ordinate system b q a

  7. Velocity and acceleration vectors Y • Position changes with time • Rate of change of r is velocity • How much is the change in a very small amount of time t r(t) r(t+t) Limit at  t0 x 0 X

  8. y v x,y,t  x Projectiles Motion of a thrown / fired object mass m under gravity Velocity components: vx=v cos  vy=v sin  Force: -mg in y direction acceleration: -g in y direction x direction y direction a: v=u+at: s=ut+0.5at2: ax=0 ay=-g vx=vcos  + axt = vcos  vy=vsin  - gt x=(vcos )t y= vtsin  -0.5gt2 This describes the motion, now we can use it to solve problems

  9. Relative Velocity 1D e.g. Alice walks forwards along a boat at 1m/s and the boat moves at 2 m/s. what is Alices’ velocity as seen by Bob ? • If Bob is on the boat it is just 1 m/s • If Bob is on the shore it is 1+2=3m/s • If Bob is on a boat passing in the opposite direction….. and the earth is spinning… • Velocity relative to an observer Relative Velocity 2D e.g. Alice walks across the boat at 1m/s. As seen on the shore: V boat 1m/s V Alice 2m/s V relative to shore

  10. Changing co-ordinate system Define the frame of reference – the co-ordinate system – in which you are measuring the relative motion. y (x’,y’) Frame S’ (boat) v boat w.r.t shore Frame S (shore) vt x’ x Equations for (stationary) Alice’s position on boat w.r.t shore i.e. the co-ordinate transformation from frame S to S’ Assuming S and S’ coincide at t=0 : Known as Gallilean transformations As we will see, these simple relations do not hold in special relativity

  11. We described the motion, position, velocity, acceleration, now look at the underlying causes Newton’s laws • First Law • A body continues in a state of rest or uniform motion unless there are forces acting on it. • No external force means no change in velocity • Second Law • A net force F acting on a body of mass m [kg] produces an acceleration a = F /m [ms-2] • Relates motion to its cause F = maunits of F: kg.m.s-2, called Newtons [N]

  12. Fb • Force exerted by block on table is Fa • Force exerted by table on block is Fb Block on table • Third Law • The force exerted by A on B is equal and opposite to the force exerted by B on A Fa Fa=-Fb Weight (a Force) (Both equal to weight) Examples of Forces weight of body from gravity (mg), - remember m is the mass, mg is the force (weight) tension, compression Friction,

  13. Tension & Compression • Tension • Pulling force - flexible or rigid • String, rope, chain and bars • Compression • Pushing force • Bars • Tension & compression act in BOTH directions. • Imagine string cut • Two equal & opposite forces – the tension mg mg mg

  14. Friction • A contact force resisting sliding • Origin is chemical forces between atoms in the two surfaces. • Static Friction (fs) • Must be overcome before an objects starts to move • Kinetic Friction (fk) • The resisting force once sliding has started • does not depend on speed N fs or fk F mg

  15. Linear Momentum Conservation • Define momentum p=mv • Newton’s 2nd law actually • So, with no external forces, momentum is conserved. • e.g. two body collision on frictionless surface in 1D Also true for net forces on groups of particles If then before m1 m2 v0 0 ms-1 Initial momentum: m1 v0 = m1v1+ m2v2 : final momentum after m1 m2 v2 v1 For 2D remember momentum is a VECTOR, must apply conservation, separately for x and y velocity components

  16. Energy Conservation • Energy can neither be created nor destroyed • Energy can be converted from one form to another • Need to consider all possible forms of energy in a system e.g: • Kinetic energy (1/2 mv2) • Potential energy (gravitational mgh, electrostatic) • Electromagnetic energy • Work done on the system • Heat (1st law of thermodynamics of Lord Kelvin) • Friction  Heat Energy measured in Joules [J]

  17. m1 m2 v2 v1 Collision revisited • We identify two types of collisions • Elastic: momentum and kinetic energy conserved • Inelastic: momentum is conserved, kinetic energy is not • Kinetic energy is transformed into other forms of energy Initial K.E.: ½m1 v02= ½ m1v12+ ½ m2v22 : final K.E. • m1>m2 • m1<m2 • m1=m2 See lecture example for cases of elastic solution Newton’s cradle

  18. Impulse Where, p1 initial momentum p2 final momentum • Change in momentum from a force acting for a short amount of time (dt) • NB: Just Newton 2nd law rewritten Q) Estimate the impulse For Greg Rusedski’s serve [150 mph]? Approximating derivative Impulse is measured in Ns. change in momentum is measured in kg m/s. since a Newton is a kg m/s2 these are equivalent

  19. Work & Energy Work is the change in energy that results from applying a force F s • Work = Force F times Distance s, units of Joules[J] • More precisely W=F.x • F,x Vectors so W=F x cos • e.g. raise a 10kg weight 2m • F=mg=10*9.8 N, • W=Fx=98*2=196 Nm=196J • The rate of doing work is the Power [Js-1Watts] • Energy can be converted into work • Electrical, chemical,Or letting the • weight fall (gravitational) • Hydro-electric power station F  x So, for constant Force mgh of water

  20. This stored energy has the potential to do work Potential Energy We are dealing with changes in energy h • choose an arbitrary 0, and look at  p.e. 0 This was gravitational p.e., another example : Stored energy in a Spring Do work on a spring to compress it or expand it Hooke’s law BUT, Force depends on extension x Work done by a variable force

  21. Work done by a variable force Consider small distance dx over which force is constant F(x) Work W=Fx dx So, total work is sum dx X 0 F Graph of F vs x, integral is area under graph work done = area dx For spring,F(x)=-kx: X x F X Stretched spring stores P.E. ½kX2

  22. Work - Energy or e.g. spring • For a system conserving K.E. + P.E., then • Conservative forces • But if a system changes energy in some other way (“dissipative forces”) • Friction changes energy to heat Then the relation no longer holds – the amount of work done will depend on the path taken against the frictional force Conservative & Dissipative Forces

  23. Simple Harmonic Motion Oscillating system that can be described by sinusoidal function Pendulum, mass on a spring, electromagnetic waves (E&B fields)… • Occurs for any system withLinear restoring Force • Same form as Hooke’s law • Hence Newton’s 2nd • Satisfied by sinusoidal expression • Substitute in to find  A is the oscillation amplitude  is the angular frequency or Frequency Hz, cycles/sec Period Sec for 1 cycle  in radians/sec

  24. SHM Examples 1) Simple Pendulum If q is small • Mass on a string q Working Horizontally: c.f. this with F=-kx on previous slide x Hence, Newton 2: mg sinq Angular frequency for simple pendulum, small deflection mg and

  25. SHM Examples2) Mass on a spring • Let weight hang on spring • Pull down by distance x • Let go! L’ Restoring Force F=-kx x In equilibrium F=-kL’=mg Energy: (assuming spring has negligible mass) potential energy of spring But total energy conserved At maximum of oscillation, when x=A and v=0 Total Similarly, for all SHM (Q. : pendulum energy?)

  26. 360o = 2 radians 180o =  radians 90o = /2 radians =t Circular Motion • Rotate in circle with constant angular speed  • R – radius of circle • s – distance moved along circumference • =t, angle  (radians) = s/R • Co-ordinates • x= R cos  = R cos t • y= R sin  = R sin t • Velocity R s y t=0 x • Acceleration N.B. similarity with S.H.M eqn 1Dprojection of a circle is SHM

  27. Magnitude and direction of motion • Velocity v=R And direction of velocity vector v Is tangential to the circle v  • Acceleration a  • a= 2R=(R)2/R=v2/R And direction of acceleration vector a • a= -2r Acceleration is towards centre of circle

  28. Angular Momentum (using v=R) • For a body moving in a circle of radius r at speed v, the angular momentum is L=r(mv) = mr2= I  The rate of change of angular momentum is • The product rF is called the torque of the Force • Work done by force is Fs =(Fr)(s/r) = Torque  angle in radians Power = rate of doing work = Torque  Angular velocity I is called moment of inertia s  r

  29. Force towards centre of circle • Particle is accelerating • So must be a Force • Accelerating towards centre of circle • So force is towards centre of circle F=ma= mv2/R in direction –r or using unit vector • Examples of central Force • Tension in a rope • Banked Corner • Gravity acting on a satellite

  30. Myth of Newton & apple. He realised gravity is universal same for planets and apples m2 m1 Gravitational Force • Any two masses m1,m2 attract each other with a gravitational force: F F r Newton’s law of Gravity Inverse square law 1/r2, r distance between masses The gravitational constant G = 6.67 x 10-11 Nm2/kg2 • Explains motion of planets, moons and tides mE=5.97x1024kg, RE=6378km Mass, radius of earth Gravity on earth’s surface Or Hence,

  31. N.B. general solution is an ellipse not a circle - planets travel in ellipses around sun Satellites • Centripetal Force provided by Gravity m R M Distance in one revolution s = 2R, in time period T, v=s/T T2R3 , Kepler’s 3rd Law • Special case of satellites – Geostationary orbit • Stay above same point on earth T=24 hours

  32.  Moment of Inertia masses m distance from rotation axis r • Have seen corresponding angular quantities for linear quantities • x; v; pL • Mass also has an equivalent: moment of Inertia, I • Linear K.E.: • Rotating body v, mI: • Or p=mv becomes: Conservation of ang. mom.: e.g. frisbee solid sphere hula-hoop pc hard disk neutron star space station R1  R2 R R

  33. Dynamics Top Five • 1D motion, 2D motion as vectors • s=ut+1/2 at2 v=u+at v2=u2+2 as • Projectiles, 2D motion analysed in components • Newton’s laws • F = ma • Conservation Laws • Energy (P.E., K.E….) and momentum • Elastic/Inelastic collisions • SHM, Circular motion • Angular momentum • L=r(mv) = mr2= I  • Moment of inertia

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