UNIT-III RIGID BODY DYNAMICS

Relation between angular momentum (L) and moment of inertia (I) The diagonal terms are called Moment of inertia terms The Non-diagonal terms are called Products of Inertia.

Principal Axes of Inertia The set of three mutually perpendicular axes having origin fixed in the rigid body, such that Products of inertia (non-diagonal terms) are zero about them and diagonal terms are non-zero. If a body possess a plane of symmetry, any line perpendicular to this plane is a principal axis.

Rotational kinetic energy of a rigid body

Euler’s Equation Consider an inertial frame S fixed in space. The torque acting on the rigid body rotating about fixed axis in the inertial frame is Consider another inertial frame S’ fixed in a rigid body, which is rotating with angular velocity in space w.r.t. frame S. The operator equation relating derivative of any vector quantity of a body rotating (S’) with some angular velocity is related to the derivative of that quantity in the coordinate system fixed in the body (S) is Hence for angular momentum, operator equation will become,

Euler’s equations

Problems with Newtonian-Galilean Transformations

Newtonian Principle of Relativity Assumption It is assumed that Newton’s laws of motion must be measured with respect to (relative to) some reference frame. If Newton’s laws are valid in one reference frame, then they are also valid in another reference frame moving at a uniform velocity relative to the first system. This is referred to as the Newtonian principle of relativity or Galilean invariance.

Inertial Frames K and K’ K is at rest and K’ is moving with velocity Axes are parallel K and K’ are said to be INERTIAL COORDINATE SYSTEMS

The Galilean Transformation For a point P In system K: P = (x, y, z, t) In system K’: P = (x’, y’, z’, t’) Parallel axes K’ has a constant relative velocity in the x-direction w.r.t. K Time (t) for all observers is a Fundamental invariant, i.e., the same for all inertial observers P x K K’ x’-axis x-axis

Problems with Newtonian- Galilean Transformation Are all the laws of Physics invariant in all inertial reference frames? For example, are the laws of electricity and magnetism the same? For this to be true Maxwell's equations must be invariant. 12 12

Problems with Newtonian- Galilean Transformation From electromagnetism we know that, Since and are constants then the speed of light is constant 13 13

Problems with Newtonian- Galilean Transformation From electromagnetism we know that, Since and are constants then the speed of light is constant However from the addition law for velocities Therefore we have a contradiction! Either the additive law for velocities and hence absolute time is wrong Or the laws of electricity and magnetism are not invariant in all frames. 14 14

Problems with Newtonian- Galilean Transformation Classical mechanics are valid at low speeds (v<<c) But are invalid at speeds close to the speed of light 15 15

The Transition to Modern Relativity Although Newton’s laws of motion had the same form under the Galilean transformation, Maxwell’s equations did not. In 1905, Albert Einstein proposed a fundamental connection between space and time and that Newton’s laws are only an approximation.

The Need for Ether Physics theories of the 19th century postulated that, just as water waves must have a medium to move across (water), and audible sound waves require a medium to move through (such as air or water), so also light waves require a medium, the "luminiferous ether or ether". Because light can travel through a vacuum, it was assumed that the vacuum must contain the medium of light. Since this medium could not be physically detected, so it was assumed to possess 1. a low density that the planets could move through it without loss of energy 2. an high elasticity to support the high velocity of light waves 3. absolute transparency as medium can not be detected physically.

Maxwell’s Equations In Maxwell’s theory the speed of light, in terms of the permeability and permittivity of free space, was given by Thus the velocity of light between moving systems must be a constant.

An Absolute Reference System Ether was proposed as an absolute reference system in which the speed of light was constant and from which other measurements could be made. The Michelson-Morley experiment was an attempt to show the existence of ether.

Earth’s motion w.r.t. ether Earth travels a tremendous distance in its orbit around the Sun, at a speed of around 30 km/s. The Sun itself is travelling about the Galactic Center at even greater speeds, and there are other motions at higher levels of the structure of the universe. Since the Earth is in motion, it was expected that the flow of ether across the Earth should produce a detectable "ether wind". Although it would be possible, in theory, for the Earth's motion to match that of the ether at one moment in time, it was not possible for the Earth to remain at rest with respect to the ether at all times, because of the variation in both the direction and the speed of the motion.

At any given point on the Earth's surface, the magnitude and direction of the wind would vary with time of day and season. By analyzing the return speed of light in different directions at various different times, it was thought to be possible to measure the motion of the Earth relative to the ether.

The Michelson–Morley experiment

Example To understand the experiment we will consider one example http://www.upscale.utoronto.ca/PVB/Harriso n/SpecRel/Flash/MichelsonMorley/Michelson Morley.html http://galileoandeinstein.physics.virginia.edu /more_stuff/flashlets/mmexpt6.htm Michelson’s great idea was to construct an exactly similar race for pulses of light, with the ether wind playing the part of the river.

The Michelson-Morley Experiment Albert Michelson (1852–1931) was the first U.S. citizen to receive the Nobel Prize for Physics (1907), and built an extremely precise device called an interferometer which is now universally called the Michelson interferometer. Principle: If we send two light signals one in the direction of motion of earth and second at its right angle to it and get these signals after reflection from two equidistant mirrors, then the difference in time taken by two signals to return to starting point can be used to determine the velocity of earth through ether.

Michelson interferometer, consists of a light source a half-silvered glass plate A (light wave on passing through this plate will be partly transmitted and partly reflect) two mirrors (M1 & M2) a telescope. One additional glass plate at B   The mirrors are placed at right angles to each other and at equal distance from the glass plate, which is obliquely oriented at an angle of 45° relative to the two mirrors.

Working 1. AC is parallel to the motion of the Earth inducing an “ether wind” 2. Light from source S is split by mirror A and travels to mirrors C and D in mutually perpendicular directions 3. After reflection the beams recombine at A slightly out of phase due to the “ether wind” as viewed by telescope E.

Assuming the Galilean Transformation D M’2 M2 l2 M1 A M’1 l1

So that the change in time is: Upon rotating the apparatus, the optical path lengths ℓ1 and ℓ2 are interchanged producing a different change in time: (note the change in denominators) using a binomial expansion, assuming v/c << 1

Results Using the Earth’s orbital speed as: V = 3 × 104 m/s together with ℓ1 ≈ ℓ2 = 11 m So that the time difference becomes Δt’ − Δt ≈ v2(ℓ1 + ℓ2)/c3 = 3.6 × 10−17 s Although a very small number, it was within the experimental range of measurement for light waves.

Result Michelson and Morley were able to measure the speed of light by looking for interference fringes between the light which had passed through the two perpendicular arms of their apparatus. These would occur since the light would travel faster along an arm if oriented in the "same" direction as the ether was moving, and slower if oriented in the opposite direction. Since the two arms were perpendicular, the only way that light would travel at the same speed in both arms and therefore arrive simultaneous at the telescope would be if the instrument were motionless with respect to the ether. If not, the crests and troughs of the light waves in the two arms would arrive and interfere slightly out of synchronization, producing a diminution of intensity. Although Michelson and Morley were expecting measuring different speeds of light in each direction, they found no discernible fringes indicating a different speed in any orientation or at any position of the Earth in its orbit around the Sun.

Typical interferometer fringe pattern expected

Michelson’s Conclusion Michelson noted that he should be able to detect a phase shift of light due to the time difference between path lengths but found none. He thus concluded that the hypothesis of the stationary ether must be incorrect. After several repeats and refinements with assistance from Edward Morley (1893-1923), again a null result. Thus, ether does not seem to exist!

Possible Explanations Ether drag hypothesis: This hypothesis suggested that the Earth somehow “dragged” the ether along as it rotates on its axis and revolves about the sun. It was proposed that velocity of earth relative to ether is zero But measurement of velocity of light near the rapidly rotating bodies showed that no appreciable velocity could be communicated to the ether Hence drag of ether with earth is not acceptable.

The Lorentz-FitzGerald Contraction Another hypothesis proposed independently by both H. A. Lorentz and G. F. FitzGerald suggested that the length ℓ1, in the direction of the motion was contracted by a factor of …thus making the path lengths equal to account for the zero phase shift. This, however, was an ad hoc assumption that could not be experimentally tested.