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Rotational Mechanics and Relativity --- Summary sheet 1
Centre of Mass
1
r dm
mi r i
For discrete masses: R 
For continuous bodies:

M body
M i
Static equilibrium: the two conditions for a body in static equilibrium are:
(1)
The vector sum of all the external forces acting on the body is zero.
(2)
The sum of all the turning moments about an axis through any point is zero.
1

F ext   F i  0
i
Circular motion
R
Gext   Gi 
i
For small angles <<1 radian, we can describe rotation
by a vector, θ , through O and along OQ, in the
direction of OQ and of magnitude equal to θ
 0.
i i
i
Q

O
We define the angular velocity vector:
 Fl
B
θ
a
s
A
and the angular acceleration vector:
Moment of force: Force F acts at point P, which is at position
vector r from point O. The moment of the force is G = Fr sin( )
perpendicular to both r and F. The moment vector G = r  F
Moment of a couple: a couple is a combination of two
equal and opposite forces which are not in line with each
other. The moment of a couple is independent of the origin.
Rotational Mechanics and Relativity --- Summary sheet 2
Angular acceleration: Rotational equivalent of acceleration
angular acceleration
.Mass m at radius r needs force
is the
.
Moment of Inertia: Rotational equivalent of the force F is the
moment
, so rotational equivalent of Newton 2 is
, where
is the moment of inertia.
Moment of inertia for continuous bodies:
Examples. Rod about end
; about centre of mass (CoM)
.
Disc about axis perpendicular to plane
(also cylinder).
Parallel axis theorem: Relates moment of inertia
about CoM to a parallel axis displaced a perpendicular
distance a from it.
Example: rod about end and about CoM (see above).
.
Perpendicular axis theorem: Relates moment of inertia
of a laminar about an axis perpendicular to the plane to the
moments in the x-y plane.
Rotational Mechanics and Relativity --- Summary sheet 3
The angular momentum, L, is the moment of the momentum about a
point. A particle at A of mass m and at position vector r relative to
point B, has momentum P = mv and angular momentum L = r  P =
mr  v .
Angular momentum is conserved in an isolated system.
Consider a system of N interacting particles.
The rate of change of L:
The value of Gint is zero since The internal interaction on the ith
particle by the jth particle is in line and oppositely directed to the
interaction on the jth particle by the ith particle by Newton 3.
In general
so, if the system is isolated and Gext = 0,
the angular momentum L is constant in time.
The impulse of a force changes the momentum:
The angular impulse changes the angular momentum
Rotational kinetic energy:
Examples: compound pendulum. Period of small oscillations:
Rotational Mechanics and Relativity --- Summary sheet 4
General motion of rigid body.
1) Centre of Mass R moves as the total mass M under sum of all external forces:
2) Rotation about CoM given by total moment of forces:
Example: cylinder rolling down plane (v=aw)
Linear momentum:
Angular momentum:
Solving:
Energy: rotational KE
Linear KE
Rotating frames. Particle rotating at radius r = (x,y),
Centripetal force
Rotational Mechanics and Relativity --- Summary sheet 5
Linear and rotational equivalents:
Gyroscopes:
Heavy flywheel under
Couple G precesses at
Angular velocity W
About z-axis.
W L sin   G
In vectors: G = Ω  L = I Ω  ω
Examples: Earth’s precession of equinoxes
Zeeman effect; Magnetic Resonance Imaging
Einstein’s theory of Special Relativity.
A frame of reference is a set of axes used to define points in space (or ‘events’ in space and
time). Cartesian frames are (x, y, z), but others are commonly used.
In spacetime an event is given by (x, y, z, t).
We need a frame of reference in which to define positions, velocities, and accelerations.
Examples: rotation of axes in space. Galilean transform between S and S’ moving at v in x:
x’=x-vt; y’=y; z’=z; t’=t. x=x’+vt’; y=y’; z=z’; t=t’. Example: train passing signals.
Rotational Mechanics and Relativity --- Summary sheet 6
Nineteenth-century physicists thought light waves must travel in a medium: the aether.
Bradley observed stellar aberration in 1725 due to motion of the Earth’s
4
orbit 3104m s1 compared to the speed of light 3108 m s1  v/c  10 .
This wasn’t evidence for the aether (relativity also predicts aberration).
Michelson and Morley set up an optical interferometer sensitive enough
to detect aether drift. A coherent light beam was divided into two
perpendicular paths which would have a time difference
No aether drift was ever found. Einstein (1905) argued that light travelled
at a finite speed
in a vacuum. He formulated two postulates:
1.
The laws of physics are the same in every inertial (un-accelerated) frame.
2.
The speed of light in a vacuum is the same for all observers.
These postulates have far-reaching implications for space and time.
Time intervals. The SI second is defined to be exactly 9,192,631,770 cycles of caesium 133.
Einstein devised a thought experiment that showed that a moving
observer’s clock ran slower than a stationary observer’s clock:
, where
He predicted relativistic time dilation. The moving observer thinks
that the stationary observer’s clock is slower by the same factor.
Rotational Mechanics and Relativity --- Summary sheet 7
Einstein’s postulates lead to the following conclusions: Time dilation:
Simultaneity: events simultaneous in one frame are not necessarily so in another.
Length contraction parallel to motion:
. No length change perpendicular to motion.
The effect is tiny in every day life: 70 mph for 6 years causes a 1 s shift.
The twin paradox: the moving twin ages more slowly than her brother on Earth.
Lorentz Transformation can be used to solve relativistic problems in a
straightforward manner. An event, A, which occurs at position
(x,y,z,t) as measured in the laboratory frame S.
The same event is also viewed in another frame
S’, moving at speed v along x, with its axes
parallel to those of S, and such that
x = x’ = 0 at t = t’ = 0 (standard configuration).
The Lorentz transformation:
This is VERY IMPORTANT.
Please learn it.
Also important are
spacetime diagrams.
Rotational Mechanics and Relativity --- Summary sheet 8
The Lorentz transformation:
This is VERY IMPORTANT.
Please learn it.
There are many example of 4-vectors in special relativity, all
of which transform using the same transformation matrix
i.e. b = A.b, where b and b are 4-vectors.
Examples: Moving and stationary twins.
Spacetime diagrams. Simultaneity example.
Experimental evidence for Special Relativity
Time dilation in the decay of muons produced by cosmic rays at the top of the atmosphere
have lifetimes of 2 s, so should travel only a few hundred metres before decaying. Their
speeds are close to c, so most of them travel through many tens of km of atmosphere due to
time dilation.
Michelson-Morley experiment. This is evidence for the absence of the Aether. Jaseja, Javan,
Murray & Townes (1964) showed that any effect is less that 0.1% of that expected.
Magnetic effects. The magnetic force between two l current-carrying wires can be calculated
from relativistic modifications of the electrostatic forces between the charges in the wires.
GPS Clocks. The rates of the clocks in the Global Satellite Positioning System satellites need to
be adjusted relative to those on the ground for both the time dilation of special relativity and
the general relativistic effect of the difference in gravitational potential.
Rotational Mechanics and Relativity --- Summary sheet 9
Addition of velocities: particle moving at
so that
Similarly
and
in S’.
.
so that
Aberration of light: Bradley measured the aberration of light, but
relativity also predicts aberration. A photon emitted from a light source L stationary in frame
S’ has a corresponding angle θ viewed from frame S given by the addition of speeds formulae:
In Bradley’s case, θ = 90°, and the direction of the photon was incoming rather
than outgoing. So sin( ) = cos( )  ,  v/c as Bradley measured.
Relativistic Doppler effect. Consider a pulsing light source at rest in a frame
S’. The time and space between two pulses measured in frame S
is
. The observer in S sees the light source
move between pulses, so the second
pulse has further to go.
Rotational Mechanics and Relativity --- Summary sheet 10
Spacetime intervals.
In 2-D Euclidean space the distance invariant is
The equivalent of Pythagoras’s theorem in Special Relativity
is
Spacetime (Minkowski) diagrams: We plot events
in spacetime, with the geometry of Minkowski
Space rather than Euclidean space. The path of a
photon is a straight line at 45° to the axes. The
vertical lines AA and BB represent the paths
Through space of stationary observers. The line
CC is the path (world line) of a moving observer,
with v < c. The scale is given by the invariant
distance x2 – c2t2 = 1.
Relativistic momentum. The conserved
momentum is
Rotational Mechanics and Relativity --- Summary sheet 11
Causality and spacetime structure:
If one event, P, causes another Q then Q must lie in the future cone
of P, as nothing can travel faster than light. The time interval
PQ is ‘time-like’, i.e. ctPQ > xPQ. And it is possible to transform
to another frame S in which P and Q both occur at the same place,
separated only by a time interval. If the interval is ‘space-like’
i.e. ct < x, as in PR, it is possible to find a frame in which P and R are
simultaneous, and yet another in which R occurs before P. It is then not
possible for R to have been caused by P. The event R lies in the ‘elsewhere’of P.
Relativistic Dynamics: Summary.
Rotational Mechanics and Relativity --- Summary sheet 12
The Energy-Momentum invariant: We think of x and ct as two components of a spacetime four-vector, and that the ‘length’ or norm of this four-vector was invariant under the
Lorentz transformation, i.e.
In the same way, p and E/c are also components of another four vector called the energymomentum four vector. This means that the components transform by the Lorentz
transformation but it also means that there is an associated invariant quantity (i.e. its ‘length’
or ‘norm’) which remains the same for a given system when viewed at any time in any
inertial frame. This is very powerful and helps to simplify problems considerably. The
invariant is E2  p2c2 = E 2  p 2c2 = m2c4 .
,
where E is the total energy, p is the total
momentum of the given system. The value of m
is the total mass in a frame (if there is one) in
which all the particles of the system are at rest.
For a system of more than one particle, use:
2
2

 
 2
  Ei     p i  c is invariant.
 i   i 
The energy-momentum 4-vector
We saw previously that the Lorentz transform for
space and time may be written as
 ct    
 
 x     v
c
 y   
   0
 z  
   0
0 0  ct 
c
 

0 0  x 
 
0
1 0  y 
z

0
0 1  
 v
ct and x, y, z form the components of a 4-vector,
and we can write the Lorentz transformation
more generally as b = A.b, where b and b are 4vectors and A is the transformation matrix which
is given by
 

 v
A
c
 0
 0

0 0 
c

0 0

0
1 0
0
0 1 
 v
It can be shown that energy and momentum form a four
vector, i.e. b = (E/c, px, py, pz):
 E c  


 px     v
c
 p   
 y   0
 p  
 z   0
0 0  E c
c



0 0  px 


p
0
1 0  y 
p 

0
0 1  z 
 v
Note
The ‘length’ or norm of a 4-vector is invariant
under the Lorentz transformation. The square
of the norm is bb which in Minkowski
geometry is given by
bb = b12  b22  b32  b42
For the E-p 4-vector this is
(E2/c2)  px2  py2  pz2
and for the t-r 4-vector it is
(c2t2)  x2  y2  z2