centre of mass - university of cambridge

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.

ii

F Fext 0 . 0i i i

i i

G FlGext

Rotational Mechanics and Relativity --- Summary sheet 1 Centre of Mass

1i i

i

mM

R r

body

1dm

M R rFor continuous bodies: For discrete masses:

θ

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

Circular motion

O

Q

A

B s

a

We define the angular velocity vector: 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 is the angular acceleration .Mass m at radius r needs force .

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


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.

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

orbit 3104m s1 compared to the speed of light 3108 m s1 .

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.


v/c 104

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.


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.

The Lorentz transformation:

This is VERY IMPORTANT.

Please learn it.

Rotational Mechanics and Relativity --- Summary sheet 9 Addition of velocities: particle moving at in S’.

so that and .

Similarly 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.

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. ctPQ > 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. ct < 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.


The Energy-Momentum invariant: We think of x and ct as two components of a space-time 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 energy-momentum 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 .

centre of mass - university of cambridge

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