Length Contraction
An object at rest in an inertial frame S is continually changing its time coordinates. It is described by an infinite succession of space-time events. A point particle at rest in an inertial frame S is described in space-time by a line made up of all the point events that have the same values of (x, y, z) in S. The two lines parallel to the t-axis, and represent x=x1 and x=x2, could be the world lines of the ends of the body whose length in S is always measured as l0.
l0=x2-x1
Derivation
Suppose now that we want to define a procedure for finding the length of the body by means of measurements made in some other frame S’. Measuring the positions of the two ends of the body at the same time t’ as judged in S’, any line representing t’=constant will intersect the world lines of the ends of the body at two points in the x-t diagram. The length l of the object as measured in S’ is given by
l=x2’-x1’
Now as measured in S these same two-point events have x coordinates equal to x1 and x2, respectively independent of the time as measured in S. Using the Lorentz transformation we have
so that the measured length of the moving rod, x2-x1 Is contracted by a factor γ from its rest length, x2’-x1’.
So what is length is measured to be greatest its rest relative to the observer so then it must be the velocity V relative to the observer its measured length is contracted in the direction of this motion by the factor
Whereas its dimensions are perpendicular to the direction of its motion and are unaffected.
You might be interested in Voyager Probes: Top 10 Amazing Facts To Make You Awestruck.
Time dilation
A Clock is measured to go at its fastest rate when it is at rest related to the observer. When it moves with velocity relative to the observer, its rate is measured to have been slowed down by a factor
Consider a clock to be at rest at the position x’ in the S’ frame. It may simplify matters to picture the hand of this clock going around and to let unit time be the time it takes the hand of the clock to go around once. Hence, the events we observe (the two successive coincidences of the hand of the clock with a given marker on the face of the clock) span the time interval t’ to t’+1 in the primed coordinates. The S frame observer records these events as occurring at times
The clock in the S’ frame is at a fixed position x’, but the times t1 and t2 are read from two different clocks in the S frame, namely the stationary S clock that happens to be coincident with the moving clock at the beginning of the interval and the stationary S clock coincident with the moving clock at the end of the interval.
You might want to see Exciting Visit of an Interstellar Object Will Make You Feel Eerie.
These clocks are synchronized, however, so that the time interval they record for the event is simply
Clearly, if, instead of unit time, the S’ clock recorded a time interval t2’-t1’, the S clock would have recorded the corresponding interval
Hence, unit time measured on the S’ clock is recorded as a longer time on the S clocks. From the point of view of observer S, the moving S’ clock appears slowed down, that is, it appears to run at a rate that is slow by a factor
This is known as time dilation. This result applies to all S’ clocks observed from S, for the location x’ in our proof was arbitrary.
Also see Betelgeuse is Surprisingly Smaller, Closer to Us.
For more content and updates, do follow us on Social Media
Join our Whatsapp Group, Facebook Group, and Telegram Group.
- Are You an AI Expert? Take Our Quiz and Find Out Now!
Test your AI knowledge with our Ultimate AI Quiz! Whether you’re a student or a professional, take the quiz and learn about this fascinating field! - THERMODYNAMICS
It is the branch of physics that deals with the concepts of heat and temperature and the inter conversion of heat and other forms of energy. - THERMAL PROPERTIES OF MATTER
Temperature is a relative measure or indication of the hotness or coldness of a body. The temperature of a body determines the direction of the flow of energy. The SI unit of temperature is kelvin (K). °C is a commonly used unit of temperature. - MECHANICAL PROPERTIES OF FLUIDS – PART 2
The viscous drag increases with increases velocity of the body, but it is found that the body after attaining certain velocity starts moving with a constant velocity called terminal velocity. - MECHANICAL PROPERTIES OF FLUIDS – PART 1
The materials that can flow are called fluids. Liquids and gases are collectively known as fluids. Unlike a solid, a fluid has no definite shape of its own.
Share this article