Lecture Notes (Classical Relativity)

Intro: - different observers may not agree upon the description of motion of an object - if you were traveling in a car at a constant speed of 70 mph, everything inside the car appears to be motionless; for example, a baseball resting in one of the passenger's hands does not appear to be moving to the other people in the car

- if the person were to drop the baseball, it would appear to all occupant of the car that the ball would fall straight down

- an observer, not in the car but standing on the side of the road, would see the ball in the car fall differently; he would see the ball fall, not straight down, but rather in a parabolic path

- this situation, where two different observers see the motion of an object differently is known as Galilean or Classical Relativity - the person in the car has one perspective and the person standing on the road watching the car go by has another perspective - a physics term is used to describe different perspectives, it is called frame of reference or reference system - a reference system is a collection of objects not moving relative to each other that can be used to describe the motion of other objects Reference Systems: - in order to describe a physical event, it is necessary to choose a frame of reference - according to the principle of Galilean relativity, the laws of mechanics must be the same in all inertial frames of reference - for example, in the situation above with the person dropping a baseball in a moving car, the forces affecting the ball would be the same for the reference frame of those inside the car as well as in the reference frame of the person standing on the

road watching the car go by (read p. 156 in text for a detailed explanation) - although you disagree with your friends' description of the ball's path, you agree on the acceleration and the forces involved; any experiments that you do in your reference system will yield the same accelerations and the same forces that your friends find in their system; in both cases the laws of motion explain the observed motion Inertial Reference Frame: - in each frame of reference above, Newton's first law of motion was valid - as a result, these reference frames are called inertial reference frames - any reference system that has a constant velocity relative to an inertial system is also an inertial system - the principle that the laws of motion are the same for any two inertial reference systems is called the Galilean principle of relativity - Galileo stated that if one were in the hold of a ship moving at a constant velocity, there would be no experiment this person could perform that would detect the motion - this means that there is no way to determine which of the two inertial reference systems is "really" at rest - there seems to be no such thing in our Universe as an absolute motion in space - there is no preferred frame of reference for describing the laws of mechanics; all motion is relative

Comparing Velocities: - although you each see different velocities, in the above case, you can at least agree that each person's observations make sense within their respective reference system - when you measure the velocity of the ball moving in the van, the value you get is equal to the vector sum of the van's velocity (measured in your system) and the ball's velocity (measured relative to the van) - suppose your friends roll the ball on the floor at 2 meters per second due east and the van is moving with a velocity of 3 meters per second due east relative to your system - in this case the vectors point in the same direction, so you simply add the speeds to obtain 5 meters per second due east

- if, instead, the ball rolls due west at 2 meters per second, you measure the ball's velocity to be 1 meter per second due east

- although this rule works well for speeds up to millions of kilometers per hour, it fails for speeds near the speed of light, about 300,000 kilometers per second (186,000 miles per second) - this is certainly not a speed that we encounter in our everyday activities; the fantastic, almost unbelievable, effects that occur at speeds approaching that of light are the subject of our next chapter (Einstein's Relativity) Non-Inertial Reference Systems: - a reference frame that is accelerating relative to another is called a non-inertial reference frame; it may also be called an accelerated reference frame - if we once again take the example of the people driving in the van we can see an example of a non-inertial reference frame

- now, the van is speeding up rather than traveling at a constant speed - when the person in the van drops the ball, instead of the ball falling straight down, the ball falls toward the back of the van

- there is an apparent force that accelerates the ball backward - this force does not arise from any physical interaction, however, and is therefore not a true force - these forces arise from the acceleration of the non-inertial reference frame and are called fictional forces or pseudo forces; your textbook calls them inertial forces - pseudo forces seem real to those objects in the non-inertial reference frame; but they are not true forces Pseudo Force Examples:

1) Accelerating In A Straight Line - when a car accelerates hard, the common human response is to feel "pushed back into the seat"; in an inertial frame of reference attached to the road, there is no physical force moving the rider backward; however, in the rider's non-inertial reference frame attached to the accelerating car, there is a backward fictitious force

2) Circular Motion - a similar effect occurs in circular motion, circular for the standpoint of an inertial frame of reference attached to the road, with the fictitious force called the centrifugal force, fictitious when seen from a non-inertial frame of reference - if a car is moving at constant speed around a circular section of road, the occupants will feel pushed outside, away from the center of the turn - if you are a person riding in a spinning amusement park ride like the one below, the riders feel a fictitious centrifugal force sticking them to the wall; the only true force is the centripetal force directed inwards; your body is trying to move in a straight line due to inertia, but the wall's centripetal force causes the riders to travel in a circle

Earth: Nearly An Inertial System: - the Earth acts almost like an inertial reference frame; that is, a frame which is traveling at constant velocity - as a result, we cannot detect that the Earth is moving at all - this is similar to closing your eyes and holding your ears on an airplane in mid-flight; you cannot tell that you are moving even though you are moving several hundred miles per hour relative to the ground - because we cannot feel the Earth moving, this has led to many scientific misconceptions over the years - up until only a few hundred years ago, people (including many brilliant scientists) believed that the Earth was stationary and the planets, stars, Sun and the Moon orbited the Earth - this was known as the geocentric model (See diagram below)

- the geocentric model was developed for many years into a sophisticated mathematical model which was very accurate at predicting the movement of celestial objects; the person credited with formalizing the geocentric theory was Ptolemy in AD 150 (See Painting Below)

- in the 1500's, a scientist from Poland found the geocentric theory convoluted and too complex to explain the motions of the planets - as a result, Copernicus revived an ancient Greek scientist's proposal that the Earth moves around the Sun and the stars remain fixed - this was initially put forth in 281 B.C. by Aristarchus, but was rejected - Copernicus said that the Earth rotated on its axis once every 24 hours, while orbiting the Sun once every year; this was known as the heliocentric model (See diagram below)

- the heliocentric model worked better than the geocentric model in correctly predicting the order and relative distances of the planets - Copernicus' model however, met resistance because it could not account for stellar parallax - since the Earth travels in a circle around the Sun in the heliocentric theory, the position of the stars should appear to change during the year

- one of the reason's it took so long to accept the heliocentric theory is that Galileo had not yet performed his work on inertial reference systems - people thought, for example, that if the Earth were moving around the Sun, that if a bird left a tree branch to fly after food, that the Earth would leave the bird far behind

- because of the fact that the Earth is nearly an inertial reference frame, it took thousands of years to convince people that the Earth actually rotates on its axis while orbiting the Sun Foucault Pendulum: - a French physicist proved that the Earth was truly rotating in the mid-1800's - Jean Bernard Leon Foucault used a large pendulum for his demonstration in the Paris Observatory - remember, a pendulum has something hanging from a fixed point which, when pulled back and released, is free to swing down by force of gravity and then out and up because of its inertia - Foucault set the pendulum in motion, and over time it appeared that the pendulum rotated across the floor

- while the pendulum seems to change its path during the day -- actually it is the floor beneath it that changes, being twisted around by the daily rotation of Earth - the pendulum has wire in tapered support which permits bending in a slightly different direction on each swing; the wire is flexible steel aircraft control cable - air resistance would normally stop the pendulum after a few hours -- so an iron collar is installed on the wire surrounded by an electromagnet that attracts the collar as the bob swings out, then shuts off automatically as it swings back, thus, keeping pendulum going; the magnet is turned on and off by a switch which is activated when the support wire interrupts a beam of light shining across its path - since it is tied to the building, the pendulum will travel laterally as the building moves laterally, but because of the way it is suspended it will not twist around if the building twists around - so, if the pendulum seems to rotate with respect to the floor, and we know there is no force available to make the pendulum rotate, then it must be the floor that is rotating, and if the floor is attached to the earth, then it must be the Earth that is rotating - if the pendulum was at the pole the floor would twist under the pendulum, the building floor would twist around the earth's axis every 24 hours

motion of a plane around a

perpendicular axis

If the Pendulum was at the POLE

the floor would twist under the pendulum -- the building floor would twist around the earth's axis every 24 hours.

motion of a plane in a circle around an

axis parallel to the plane

If the Pendulum was at the EQUATOR

the building floor would not twist at all but the building would travel eastward on the earth's axis

BUT -- it is clear that the pendulum --which doesn't twist-- would stay in its original plane.

--and the pendulum--being tied to the building -- would travel right along with the building with no visible effect -- since there is no twisting motion.

Coriolis Effect: - the Coriolis effect is caused by the Coriolis force; a pseudo-force that occurs when you move along a radius of a rotating system

- the Coriolis force is named after Gaspard-Gustave Coriolis, a French scientist who described it in 1835 - one of the most notable examples is the deflection of winds moving along the surface of the Earth to the right of the direction of travel in the Northern hemisphere and to the left of the direction of travel in the Southern hemisphere - this effect is caused by the rotation of the Earth and is responsible for the direction of the rotation of large cyclones: winds around the center of a cyclone rotate counterclockwise on the northern hemisphere and clockwise on the southern hemisphere