P1X Dynamics & Relativity : Newton & Einstein

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 – Newton’s Laws Simple Harmonic Motion Circular Motion READ the textbook! section numbers in syllabus http://ppewww.ph.gla.ac.uk/~parkes/teaching/Dynamics/Dynamics.html Chris Parkes October 2007

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) Limit at  t0 r(t+t) 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 2m/s. What is Alice’s 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 2m/s θ V Alice 1m/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. Force Components • Force is a Vector • Resultant from vector sum • Resolve into perpendicular components 

14. Free Body Diagram • Apply Newton’s laws to particular body • Only forces acting on the body matter • Net Force • Separate problem into each body e.g. Body 1 Body 2 SupportingForce from plane Tension in rope Tension In rope (normal force) Block Weight Friction Block weight

15. 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

16. 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

17. 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

18. SHM General Form  Phase (offset of sine wave in time) Displacement Oscillation frequency A is the oscillation amplitude - Maximum displacement

19. SHM Examples1) 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?)

20. q L x mg sinq mg SHM Examples 2) Simple Pendulum • Mass on a string Working along swing: Not actually SHM, proportional to sin, not  but if q is small c.f. this with F=-kx on previous slide Hence, Newton 2: and Angular frequency for simple pendulum, small deflection

21. 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

22. 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

23. 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

24. 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,

25. 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

26. Dynamics I – Key Points • 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 • Action & reaction • SHM • Circular motion (R,) Oscillating system that can be described by sinusoidal function Force towards centre of circle