Moving Coordinate Systems and Special Relativity

Appendices

Summary

• For the description of motions one needs a coordinate system. Coordinate systems in which the Newtonian Laws can be formulated in the form, discussed in Sect. 2.6 arte called inertial systems. Each coordinate system which moves with constant velocity v against another inertial system is also an inertial system.

• The transformation of coordinates $(x,y,z)$, of time t and of velocity v and therefore also of the equation of motion from one to another inertial system is described by the Lorentz transformations. They are based on the constancy of the speed of light c, confirmed by experiments, which is independent of the chosen inertial system and has the same value in all inertial systems. For small velocities $v\ll c$ the Lorentz transformations approach the classical Galilei transformations.

• The description of motions in accelerated systems demand additional accelerations, which are caused by „inertial or virtual“ forces. In a rotating system with constant angular velocity these are the Coriolis force $\boldsymbol{F}_{\mathrm{C}}=2\,\mathrm{m}(\boldsymbol{v}^{\prime}\times\boldsymbol{\omega})$ which depends on the velocity $v^{\prime}$ of a body relative to the rotating system, and the centrifugal force $\boldsymbol{F}_{\mathrm{cf}}=m\cdot\boldsymbol{\omega}\times(\boldsymbol{r}\times\boldsymbol{\omega})$ which is independent of $v^{\prime}$.

• The theory of special relativity is based on the Lorentz transformations and discusses the physical effects following from these equations when the motion of a body is described in two different inertial systems which move against each other with constant velocity v. An essential point is the correct definition of simultaneity of two events. Many statements of special relativity can be illustrated by space-time diagrams $(x,ct)$ (Minkowski diagrams), as for instance the length-contraction or the time-dilatation. Such diagrams show that these effects are relative and symmetric, which means that each observers measures the lengths in a system moving against his system contracted and the time prolonged. The description of the two observers O and $O^{\prime}$ are different but consistent. There is no contradiction.

• For the twin-paradox an asymmetry occurs, because the astronaut $\mathrm{A}$ changes its inertial system at the point of return. It is therefore possible to attribute the time dilation unambiguously to one of the observers.

• The statements of special relativity have been fully confirmed by numerous experiments.

Problems

3.1

An elevator with a cabin heights of $2.50\,\mathrm{m}$ is accelerated with constant acceleration $a=-1\,\mathrm{m}/\mathrm{s}^{2}$ starting with v = 0 at t = 0. After $3\,\mathrm{s}$ a ball is released from the ceiling.

• a) At which time reaches it the bottom of the cabin?

• b) Which distance in the resting system of the elevator well has the ball passed?

• c) Which velocity has the ball at the time of the bounce with the bottom in the system of the cabin and in the system of the elevator well?

3.2

From a point $\mathrm{A}$ on the earth equator a bullet is shot in horizontal direction with the velocity $v=200\,\mathrm{m}/\mathrm{s}$.

• a) in the north direction

• b) in the north-east direction $45^{\circ}$ against the equator

• c) In the north-west direction $135^{\circ}$ against the equator

What are the trajectories in the three cases described in the system of the rotating earth?

3.3

A ball hanging on a $10\,\mathrm{m}$ long string is deflected from its vertical position and rotates around the vertical axis with $\omega=2\pi\cdot 0.2\,\mathrm{s}^{-1}$. What is the angle of the string against the vertical and what is the velocity v of the ball?

3.4

In the edge region of a typhoon over Japan (geographical latitude $\varphi=40^{\circ}$) the horizontally circulating air has a velocity of $120\,\mathrm{k}\mathrm{m}/\mathrm{h}$. What is the radius of curvature r of the path of the air in this region?

3.5

A fast train $(m=3\cdot 10^{6}\,\mathrm{k}\mathrm{g})$ drives from Cologne to Basel with a velocity of $v=200\,\mathrm{k}\mathrm{m}/\mathrm{h}$ exactly in north-south direction passing $48^{\circ}$ latitude. How large is the Coriolis force acting on the rail? Into which direction is it acting?

3.6

A body with mass $m=5\,\mathrm{k}\mathrm{g}$ is connected to a string with $L=1\,\mathrm{m}$ and rotates

• a) in a horizontal plane around a vertical axis

• b) in a vertical plane around a horizontal axis

At which angular velocity breaks the string in the cases a) and b) when the maximum tension force of the string is 1000 $\mathrm{N}$?

3.7

A plane disc rotates with a constant angular velocity $\omega=2\pi\cdot 10\,\mathrm{s}^{-1}$ around an axis through the centre of the disc perpendicular to the disc plane. At time t = 0 a ball is launched with the velocity $v=\{v_{r},v_{\varphi}\}$ with $v_{r}=10\,\mathrm{m}/\mathrm{s}$, $v_{\varphi}=5\,\mathrm{m}/\mathrm{s}$ (measured in the resting system) starting from the point $\mathrm{A}$ $(r=0.1\,\mathrm{m},\varphi=0^{\circ})$. At which point $(r,\varphi)$ does the ball reach the edge of the disc?

3.8

A bullet with mass $m=1\,\mathrm{k}\mathrm{g}$ is shot with the velocity $v=7\,\mathrm{k}\mathrm{m}/\mathrm{s}$ from a point $\mathrm{A}$ on the earth surface with the geographical latitude $\varphi=45^{\circ}$ into the east direction. How large are centrifugal and Coriolis force directly after the launch? At which latitude is its impact?

3.9

Two inertial systems S and $S^{\prime}$ move against each other with the velocity $v=v_{x}=c/3$. A body $\mathrm{A}$ moves in the system S with the velocity $\boldsymbol{u}=\{u_{x}=0.5c,u_{y}=0.1c,u_{z}=0\}$. What is the velocity vector $\boldsymbol{u}^{\prime}$ in the system $S^{\prime}$ when using

• a) the Galilei transformations and

• b) the Lorentz transformations?

How large is the error of a) compared to b)?

3.10

A meter scale moves with the velocity $v=2.8\cdot 10^{8}\,\mathrm{m}/\mathrm{s}$ passing an observer $\mathrm{B}$ at rest. Which length is $\mathrm{B}$ measuring?

3.11

A space ship flies with constant velocity v to the planet Neptune and reaches Neptune at its closest approach to earth. How large must be the velocity v if the travel time, measured by the astronaut is 1 day? How long is then the travel time measured by an observer on earth?

3.12

Light pulses are sent simultaneously from the two endpoints $\mathrm{A}$ and $\mathrm{B}$ of a rod at rest. Where should an observer O sit in order to receive the pulses simultaneously? Is the answer different when $\mathrm{A}$, $\mathrm{B}$ and O moves with the constant velocity v? At which point in the system S an observer $O^{\prime}$ moving with a velocity v _x against S receives the pulses simultaneously if he knows that the pulses has been sent in the system S simultaneously from $\mathrm{A}$ and $\mathrm{B}$?

3.13

At January 1st 2010 the astronaut $\mathrm{A}$ starts with the constant velocity $v=0.8c$ to our next star α-Centauri, with a distance of 4 light years from earth. After arriving at the star, $\mathrm{A}$ immediately returns and flies back with $v=0.8c$ and reaches the earth according to the measurement of $\mathrm{B}$ on earth at the 1st of January 2020. $\mathrm{A}$ and $\mathrm{B}$ had agreed to send a signal on each New Year’s Day. Show that $\mathrm{B}$ sends 10 signals, but $\mathrm{A}$ only 6. How many signals does $\mathrm{A}$ receive on his outbound trip and how many on his return trip?

3.14

Astronaut $\mathrm{A}$ starts at t = 0 his trip to the star Sirius (distance 8.61 light years) with the velocity $v_{1}=0.8c$. One year later $\mathrm{B}$ starts with the velocity $v_{2}=0.9c$ to the same star. At which time does $\mathrm{B}$ overtake $\mathrm{A}$, measured

• a) in the system of $\mathrm{A}$,

• b) of $\mathrm{B}$ and

• c) of an observer $\mathrm{C}$ who stayed at home?

At which distance from $\mathrm{C}$ measured in the system of $\mathrm{C}$ does this occur?