P1XDynamics

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P1XDynamics RelativityNewton 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
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Motion
x
e.g

• Position m
• Velocity ms-1
• Rate of change of position
• Acceleration ms-2
• Rate of change of velocity

dx
0
t
dt
v
0
t
a
0
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Equations of motion in 1D

• Initially (t0) at x0
• Initial velocity u,
• acceleration a,

sut1/2 at2, where s is displacement from
initial position vuat
Differentiate w.r.t. time
v2u22 as
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2D motion vector quantities
Scalar 1 number Vector magnitude direction,
gt1 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
• xr cos ?, yr sin ?
• r2x2y2, tan ? y/x

Y
r
y
?
x
0
X
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vector addition

• cab
• cx ax bx
• cy ay by

y
b
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
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Velocity and acceleration vectors

• Position changes with time
• Rate of change of r is velocity
• How much is the change in a very small amount of
  time ?t

Y
r(t)
r(t?t)
Limit at ? t?0
x
0
X
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Projectiles
Motion of a thrown / fired object mass m under
gravity
Velocity components vxv cos ? vyv sin ?

• Force -mg in y direction
• acceleration -g in y direction

x direction
y direction
a vuat sut0.5at2
ax0
ay-g
vxvcos ? axt vcos ?
vyvsin ? - gt
x(vcos ?)t
y vtsin ? -0.5gt2
This describes the motion, now we can use it to
solve problems
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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 123m/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
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Changing co-ordinate system
Define the frame of reference the co-ordinate
system in which you are measuring the relative
motion.
y
Frame S (boat)
(x,y)
v boat w.r.t shore
Frame S (shore)
vt
x
x
Equations for (stationary) Alices position on
boat w.r.t shore i.e. the co-ordinate
transformation from frame S to S Assuming S and
S coincide at t0
Known as Gallilean transformations As we will
see, these simple relations do not hold in
special relativity
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Newtons laws
We described the motion, position, velocity,
acceleration, now look at the underlying causes

• 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 ma units of F kg.m.s-2, called Newtons
  N

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• Third Law
• The force exerted by A on B is equal and opposite
  to the force exerted by B on A

Fb

• Force exerted by block on table is Fa
• Force exerted by table on block is Fb

Block on table
Fa
Weight (a Force)
Fa-Fb
(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,
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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
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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
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Linear Momentum Conservation

• Define momentum pmv
• Newtons 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
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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
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Collision revisited
m1
m2
v2
v1

• 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.

• m1gtm2
• m1ltm2
• m1m2

See lecture example for cases of elastic solution
Newtons cradle
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Impulse

• Change in momentum from a force acting for a
  short amount of time (dt)
• NB Just Newton 2nd law rewritten

Where, p1 initial momentum p2 final momentum
Q) Estimate the impulse For Greg Rusedskis
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
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Work Energy
Work is the change in energy that results from
applying a force

• Work Force F times Distance s, units of
  JoulesJ
• More precisely WF.x
• F,x Vectors so WF x cos?
• e.g. raise a 10kg weight 2m
• Fmg109.8 N,
• WFx982196 Nm196J
• 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
s
F
?
x
So, for constant Force
mgh of water
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This stored energy has the potential to do work
Potential Energy
We are dealing with changes in energy
h
0

• choose an arbitrary 0, and look at ? p.e.

This was gravitational p.e., another example
Stored energy in a Spring
Do work on a spring to compress it or expand it
Hookes law
BUT, Force depends on extension x
Work done by a variable force
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Work done by a variable force
Consider small distance dx over which force is
constant
F(x)
Work WFx 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
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Work - Energy
or
e.g. spring
Conservative Dissipative Forces

• 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

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Simple Harmonic Motion
Oscillating system that can be described by
sinusoidal function Pendulum, mass on a spring,
electromagnetic waves (EB fields)

• Occurs for any system with Linear restoring Force
• Same form as Hookes law
• Hence Newtons 2nd
• Satisfied by sinusoidal expression
• Substitute in to find ?

A is the oscillation amplitude ? is the angular
frequency
or
Period Sec for 1 cycle
Frequency Hz, cycles/sec
? in radians/sec
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SHM Examples
1) Simple Pendulum

• Mass on a string

If q is small
Working Horizontally
q
c.f. this with F-kx on previous slide
x
Hence, Newton 2
mg sinq
Angular frequency for simple pendulum, small
deflection
mg
and
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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-kLmg
Energy
(assuming spring has negligible mass)
potential energy of spring
But total energy conserved
At maximum of oscillation, when xA and v0
Total
Similarly, for all SHM (Q. pendulum energy?)
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Circular Motion
360o 2? radians 180o ? radians 90o ?/2
radians

• 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
t0
x

• Acceleration

N.B. similarity with S.H.M eqn 1D projection of
a circle is SHM
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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/Rv2/R
And direction of acceleration vector a
a -?2r
?Acceleration is towards centre of circle
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Angular Momentum

• For a body moving in a circle of radius r at
  speed v, the angular momentum is
• Lr ?(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

(using v?R)
I is called moment of inertia
s
?
r
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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
• Fma mv2/R in direction r
• or using unit vector
• Examples of central Force
• Tension in a rope
• Banked Corner
• Gravity acting on a satellite

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Gravitational Force
Myth of Newton apple. He realised gravity
is universal same for planets and apples

• Any two masses m1,m2 attract each other with a
  gravitational force

F
F
r
Newtons 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

mE5.97x1024kg, RE6378km Mass, radius of earth
Gravity on earths surface
Or
Hence,
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Satellites
N.B. general solution is an ellipse not a circle
- planets travel in ellipses around sun

• Centripetal Force provided by Gravity

m
R
M
Distance in one revolution s 2?R, in time
period T, vs/T
T2?R3 , Keplers 3rd Law

• Special case of satellites Geostationary orbit
• Stay above same point on earth T24 hours

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Moment of Inertia

• 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 pmv becomes
• Conservation of ang. mom.
• e.g. frisbee solid sphere hula-hoop
• pc hard disk neutron star space station

masses m
distance
from
rotation axis r
R1
?
R2
R
R
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Dynamics Top Five

• 1D motion, 2D motion as vectors
• sut1/2 at2 vuat v2u22 as
• Projectiles, 2D motion analysed in components
• Newtons laws
• F ma
• Conservation Laws
• Energy (P.E., K.E.) and momentum
• Elastic/Inelastic collisions
• SHM, Circular motion
• Angular momentum
• Lr ?(mv) mr2? I ?
• Moment of inertia