KNUST PHY 170 Physics For Chemical Engineers 2022 PDF

Kwame Nkrumah University of Science & Technology, Kumasi, Ghana KNUST PHY 170 PHYSICS FOR CHEMICAL ENGINEERS July, 2022  Kwame Nkrumah University of Science & Technology, Kumasi, Ghana SPECIAL THEORY OF RELATIVITY ...

Kwame Nkrumah University of Science & Technology, Kumasi, Ghana KNUST PHY 170 PHYSICS FOR CHEMICAL ENGINEERS July, 2022  Kwame Nkrumah University of Science & Technology, Kumasi, Ghana SPECIAL THEORY OF RELATIVITY  Background ❑ Relativity has a reputation, among those who have not studied it, as a difficult subject. ❑ It is not the mathematical complexity that stands in the way of understanding; if you can solve a quadratic equation, you are overqualified. ❑ The difficulty lies entirely with the fact that relativity forces us to re-examine critically our ideas of space and time. ❑ At its core, relativity is about how different people perceive the same thing. www.knust.edu.gh  Background ❑ Let me start with something a situation that will make it easier to understand. ❑ Two people. One in a car and the other standing by the roadside. They both think they are at the center of the universe. www.knust.edu.gh  Background ❑ The guy in the car thinks everything around him is stationary and even the points ahead and behind him are stationary. ❑ And this is how he will look at the world around him. www.knust.edu.gh  Background ❑ What about the guy by the roadside? ❑ He also thinks he is at the center of the world so the two of them can only agree at one point. ❑ Equation can be written for the movement of the car guy with respect to the guy by the roadside. www.knust.edu.gh  Background ❑ So we can always find a relationship relating the two observation points. ❑ This is called the Galilean Transformation www.knust.edu.gh  Background ❑ So we can always find a relationship relating the two observation points. ❑ This is called the Galilean Transformation ❑ We can have the same transformation for time. www.knust.edu.gh  Background TIME ❖ What is Time? ❖ Do we really understand time? ❖ Is it fixed for everybody? ❖ Does it vary based on where you are observing it from or where you are? https://www.istockphoto.com/vector/timepiec e-gm124679470-6061815 www.knust.edu.gh  Background Albert Einstein ❖ The time was 1905. ❖ At that time he was a Patent Examiner in the Swiss Patent Office. ❖ This brilliant young man came up with a theory that will show how space and time are linked for objects that are moving at a consistent speed in a straight line. https://www.dpma.de/english/ our_office/publications/milesto ❖ This theory is known as the nes/einsteins140geburtstag/in Special Theory of Relativity. dex.html#:~:text=Einstein%20w ❖ One of its most famous aspects orked%20at%20the%20Swiss,th concerns objects moving at the e%20best...). speed of light. www.knust.edu.gh  Background ❖ Let’s say two people are sitting in a train moving relative to the environment. ❖ How will the two people sitting in the train observe each other in terms of motion? www.knust.edu.gh  Background ❖ Let’s say one person gets down and observes the train from the side of the track. ❖ Now what happens? ❖ Who is stationary and who is moving? www.knust.edu.gh  Background ❖ This is called relative motion. ❖ And we experience it everyday. ❖ Everything experiences relative motion except light. https://www.shutterstock.com/search/back+li ght www.knust.edu.gh  Background ❖ So let’s say we mount a light clock on a rocket and fire it through space and someone on Earth holds the same kind of clock. ❖ What’s the behavior of the light clock on earth and what will be the behavior in space? ❖ Looks like the light is covering a longer distance, isn’t it? ❖ Which will mean the speed of light should be comparable to the speed of the space ship. www.knust.edu.gh  Background ❖ But the Michelson-Morley experiment had shown in the 1800s that the speed of light was constant everywhere in space. And this constant is “c” from the word “Cileritas”. Meaning: speed, quickness or Speed of light ❖ So if the speed of light on Earth and in Space are the same then something else must be changing. www.knust.edu.gh  Background 𝑠 = 𝑑/𝑡 s speed on earth 𝑆 = 𝐷/𝑡 S speed in space 𝑆=𝑠 Time must be changing then. It turns out time slows down as you approach the speed of light. The person in the rocket ship will therefore experience slower time compared to the person on earth. This is the basis of the special theory of relativity and this theory proves that time is different for different observers. www.knust.edu.gh Background www.knust.edu.gh Background Gargantua and Time Dilation Miller’s planet is orbiting a supermassive black hole called Gargantua, which is shown to warp space-time so magnificently that they can see its entire accretion disk, all at once. The unique way it appears in the film was derived from equations that physicist Kip Thorne generated. Something else Kip Thorne figured out is that, given how close Miller’s planet is to Gargantua, for every hour spent on Miller’s planet, seven years will pass on Earth. www.knust.edu.gh Questions? www.knust.edu.gh www.knust.edu.gh www.knust.edu.gh www.knust.edu.gh www.knust.edu.gh www.knust.edu.gh www.knust.edu.gh www.knust.edu.gh www.knust.edu.gh www.knust.edu.gh www.knust.edu.gh www.knust.edu.gh www.knust.edu.gh www.knust.edu.gh www.knust.edu.gh www.knust.edu.gh www.knust.edu.gh www.knust.edu.gh www.knust.edu.gh www.knust.edu.gh Noticing that they arrive at the same moment and that they come from places equal distances away, the observer will decide that the two events happened simultaneous. www.knust.edu.gh What is the new observer to make of this? For the new observer, the light from A must cover a greater distance to catch up with the receding midpoint; and the light from B must cover a lesser distance to arrive at the midpoint rushing towards it. So if the two arrive at the same moment, the light from A must have left earlier than the light from B to give it greater time to cover the greater distance to get to the midpoint. That is, the flash at A happened earlier than the flash at B. The two events were not simultaneous, according to the new observer. www.knust.edu.gh www.knust.edu.gh www.knust.edu.gh Lorentz Transformation Thought Experiment Let us consider a propagating light source measured by two different observers from the x and x´ coordinate systems. Assumptions: t = 0 t − t = 0 www.knust.edu.gh Lorentz Transformation www.knust.edu.gh Lorentz Transformation www.knust.edu.gh Lorentz Transformation www.knust.edu.gh Plot results www.knust.edu.gh Plot results www.knust.edu.gh Time Dilation www.knust.edu.gh Moving clocks run slower www.knust.edu.gh  Time Dilation ❑For example, suppose that a rocket ship is moving through space at a speed of 0.8c. ❑According to an observer on earth 1.67 years pass for each year that passes for the rocket man, because for this velocity gamma=1.67 ❑But wait a second! According to the person on the rocket ship, the earth-man is moving at 0.8c. The rocket man will therefore observe the earth clock as running slow. ❑Each sees the other’s clock as running slow. www.knust.edu.gh  The Twin Paradox To bring this issue into focus, consider the following story: ❑Jane and Sally are identical twins. When they are both age 35, Sally travels in a rocket to a star 20 light years away at v=0.99c and then returns to Earth. The trip takes 40 years according to Jane and when Sally gets back, Jane has aged 40 years and is now 75 years old. Since gamma=7.09, Sally has aged only 5 years 8 months and is therefore only 40 years and 8 months old. Yet according to the above, when Sally was moving, she would see Jane’s clock as running slow. How is this possible??? www.knust.edu.gh The Twin Paradox www.knust.edu.gh Unlike the Twin Paradox, time dilation is not a thought experiment or a hypothetical concept–– it is real: www.knust.edu.gh https://www.thevintagenews.com/2016/09/16/ hafele-keating-experiment-two-atomic-clocks- flew-twice-around-world-eastward-westward- back-home-showed-different-times/ The Hafele–Keating experiment was a test of the theory of relativity. In October 1971, Joseph C. Hafele, a physicist, and Richard E. Keating, an astronomer, took four cesium-beam atomic clocks aboard commercial airliners. They flew twice around the world, first eastward, then westward, and compared the clocks against others that remained at the United States Naval Observatory. When reunited, the three sets of clocks were found to disagree with one another, and their differences were consistent with the predictions of special and general relativity. www.knust.edu.gh https://aapt.scitation.org/doi/10.1119/1.19695 08 Fast-moving muon particles take longer to decay. www.knust.edu.gh  Length Contraction ❑ The length of an object measured in a reference frame in which the object is at rest is called the proper length L0. ❑ The length L measured in a reference in which the body is moving is always shorter than L0. ❑ The length decrease between L0 and L is called the length contraction. www.knust.edu.gh Length Contraction www.knust.edu.gh Length Contraction www.knust.edu.gh Length Contraction www.knust.edu.gh  Questions Which of the following was a consequence of the Einstein Special Theory of Relativity? ❑ Events which are simultaneous to a stationary observer are simultaneous to a moving observer. ❑ Nothing can move faster than c, the speed of light in vacuum. ❑ A stationary observer will see a moving clock running at the same rate. ❑ A moving object will be stretched along its direction of motion. ❑ All of the above are true. www.knust.edu.gh WORKED EXAMPLES www.knust.edu.gh WORKED EXAMPLES www.knust.edu.gh WORKED EXAMPLES www.knust.edu.gh SOLUTIONS www.knust.edu.gh SOLUTIONS www.knust.edu.gh SOLUTIONS www.knust.edu.gh SOLUTIONS www.knust.edu.gh Kwame Nkrumah University of Science & Technology, Kumasi, Ghana 69