Chapter 26 Lecture Notes

Transcription

1 Chapter 26 Leture Notes Physis Strauss Formulas: t = t0 1 v L = L0 1 v m = m0 1 v E = m KE = m 2 KE = m 2 -m 0 2 mv 0 p= mv = 1 v E 2 = p m v + u u = vu There were two revolutions in physis in the twentieth entury. One of them is quantum mehanis. Quantum mehanis has its major impat when dealing with objets on a very small sale. The other is speial relativity. Speial relativity has its major impat when veloities are very fast. 1. REFERENCE FRAMES The theory of speial relativity deals with how two different observers in different referene frames would view the same events. The referene frame is the point of view of the observer. For instane, if I threw a ball up on a train, it would appear to go straight up and ome straight down. But to an observer who wasn t on the train, it would appear to follow a paraboli path. Speial relativity only deals with observers who are in an inertial referene frame. An inertial referene frame is a referene frame that is not aelerating. It may be moving, but it an not be aelerating. So riding in a ar at a steady veloity is an inertial referene frame, but sitting on a merry-go-round is not beause as you are sitting, you are experiening entripetal aeleration. 2. POSTULATES OF SPECIAL RELATIVITY The speial theory of relativity is, in some sense, a misnomer, for this theory is based on two absolute priniples. Beause these two priniples are ompletely absolute, there are other things that were one onsidered absolute, but are now 1

2 relative, like length, time, mass, momentum, and energy. The two absolute priniples are the two postulates of the speial theory of relativity. They are 1. The laws of physis are the same in all inertial referene frames. There is no preferred, or absolute, inertial referene frame. The laws themselves are absolute. 2. The speed of light in a vauum is always measured to be the same value in any inertial referene frame, independent of the speed of the soure or the observer. So the laws of physis, and the speed of light in a vauum are absolutes. Length, time, mass, momentum, and energy are dependent on the referene frame of the observer. Therefore, it is a postulate of the speial theory of relativity, that if I am in a spaeship traveling at the speed of light, and I turn on my headlights, the light still moves away from me at the speed of light. 3. CONSEQUENCES OF SPECIAL RELATIVITY 3.1 Simultaneity What does it mean for two events to happen simultaneously? Suppose that I am standing at the 50 yard line of a football stadium, and I notie that two people on opposite goal lines ath a pass at the same time. I onlude that the two events must have happened at the same time sine I am equal distane from eah goal line and I know that t = x/v = x/. The two events were simultaneous. Now suppose that I see someone at one goal line ath a pass before someone at the other goal line. I onlude that the events were not simultaneous sine I am the same distane from the two events, and I see one happen before the other. Now I do the same experiment but instead of standing on the 50 yard line, I am standing in the middle of a railroad ar. As one railroad ar on one train passes another railroad ar on a different train lightning strikes eah end of the ar simultaneously, from my perspetive. That is, the light from eah of the lightning strikes reahes my eyes at the same time. However, beause a person on the other train is moving relative to me, the light from one of the lightning strikes reahes him before the light from the other lightning strike. (See figure 26-6). He onludes that the events were not simultaneous sine the light from one event reahes him before the light from the other event, and they both ourred at equal distanes from him. So I think the events were simultaneous, and he thinks they were not. We see that the idea of whether two events were simultaneous or not does not have an absolute answer in speial relativity. 2

3 3.2 Time Dilation In fat, time itself does not have an absolute meaning in speial relativity. Suppose I am sitting on a moving roket ship and put a light soure on the floor. I then boune the light off of a mirror on the eiling whih is a distane D from the floor. From my perspetive it took the light the amount of time t 0 = 2D/ to go that distane. Now suppose someone is not on the roket ship is wathing the light boune off the mirror. To a person who is not in the roket ship, the roket ship appears to be moving at a veloity of v. This person will say that the light did not go a distane D, but instead went a distane 2 D + L = 2 2 D + v t 4sine L = v t/2. So the time for that person is 2 t = 2 D + v t 4/, and squaring both sides, dividing by ( t) 2, and solving for t gives ( t) 2 = (4D 2 + v 2 t 2 )/ 2 1 = (4D 2 / t 2 + v 2 )/ 2 t 2 = 4D 2 /{ 2 (1-v 2 / 2 )} t = 2D/{ 1 v } Beause t 0 = 2D/, this beomes t = t 0 / 1 v where t is the time measured by an observer in a referene frame where the two events do not our at the same loation and t 0 is alled the proper time and is the time measured by an observer in a referene frame where the two events do our in the same loation. (Whenever we see the word proper it means in a referene frame where the two events appear to happen in the same plae.) Note that t is greater than t 0 so the phenomena is known as time dilation. This states that aording to a stationary observer, a moving lok runs more slowly than an idential stationary lok. PROBLEM: An astronaut on the earth has a heartbeat rate of 70 beats/min. When the astronaut is traveling in a spaeship at 0.90, what will be the rate of his heart (a) measured by an observer in the spaeship, and (b) measured by an observer on the earth. Note that if the speed is relatively slow, even 100,000 miles/hour (59600 m/s = ). (This speed would take one to the moon in about 2.5 hours), then minute beomes minutes. These effets do not beome 3

4 notieable until one gets VERY lose to the speed of light. Even at a speed of 10% of the speed of light, m/s, then 1 minute beomes 1 minute and seonds. However, time dilation has been shown to be true using either atomi loks at relatively slow speeds, or elementary partiles moving very lose to the speed of light. The key to all of relativity will be this relationship 1/ 1 v whih is often given the symbol γ. At 2 ten-thousandths of the speed of light (59600 m/s) we find that γ is only and that normal quantities we measure do not hange muh. When the veloity of an objet is as high as one hundredth of the speed of light γ is still lose to 1. It is only When the veloity is 1 tenth of the speed of light γ is still only (So even at 1/10 the speed of light, loks only slow down by 0.5%). However, if the veloity is 95% of the speed of light, then γ is (so the time measured would inrease by a fator of 3.2). This quantity (γ) beomes large only when the veloity is very lose to the speed of light. 3.3 Length Contration If time is relative, but the speed of light is an absolute, then one would expet that distane (whih is speed times time) is also relative. Indeed, it is. When two referene frames are moving at a speed of v relative to eah other we find that v = L/ t = L 0 / t 0 L = L 0 1 v So length is not absolute. We find that aording to a stationary observer, a moving objet is shorter than an idential stationary objet. This is alled length ontration. We find that length ontration only works in the diretion of the relative veloity of the two referene frames. Again, L 0 is the length of the objet in a referene frame that is stationary with respet to the objet and L is the length of the same objet when viewed from a referene frame that is moving a veloity v with respet to the first referene frame. PROBLEM: A UFO moves by an observer at a speed of If the observer measures the UFO to be 150 m long along the diretion of motion and 50 m high, how big is the UFO aording to the pilot? 3.4 Mass Inrease It an also be shown that the mass of an objet inreases as its relative veloity inreases. We find that 4

5 m = m 0 / 1 v. PROBLEM: Suppose I throw a ball with a mass of 0.50 kg at a speed of What is the mass of the ball in flight? 4. RELATIVISTIC MOMENTUM Sine momentum is defined as mass time veloity, we find that the true momentum of an objet an only be alulated using its relativisti mass. That is, p = mv = m 0 v/ 1 v. Of ourse, unless the veloity is extremely great, then m m 0 and these two equations are the same. As an objet s speed inreases its mass inreases. Sine the fore it takes to aelerate an objet is F = ma, as its mass inreases, the fore it takes to aelerate the objet even more also inreases. As v approahes, the mass gets loser and loser to infinity. For v to atually reah, the mass would be infinite, and the orresponding fore required to reah that speed would have to be an infinite amount of fore. Consequently, nothing with mass an ever atually reah the speed of light. Only objets without mass an reah the speed of light. Consequently, photons, whih are light themselves, and therefore, move at the speed of light, must be massless partiles. 5. RELATIVISTIC MASS AND ENERGY In relativity, Einstein showed that the kineti energy of an objet is given by KE = m 2 - m 0 2. The total energy of the objet is given by E = KE + m 0 2 = m 2 =m 0 2 / 1 v This equation for energy is always orret. Even when an objet has no kineti energy it still has an energy equal to m 0 2 whih is alled the rest energy of an objet. It is the energy that the objet has even when it is not moving. We have seen that it is this rest energy (the energy inherent in the mass of the partile) whih is released during nulear reations. We an rewrite this equation for energy in the following way (the derivation is on page 765). E 2 = m p 2 2 5

6 This equation is widely used for the total energy. Like the equation for energy above, both of these equations are valid regardless of the speed of the objet. Page 764 shows that for low speeds the equation E = KE + m 0 2 = m 0 2 / 1 v redues to the more familiar equation KE = (1/2)m 0 v 2. PROBLEM: (a) What is the total energy and the kineti energy of an eletron moving at 0.850? (b) What would the kineti energy be if relativity was not onsidered? The rest energy of an eletron is MeV. (Remember that when the rest energy is given in eletron volts, it has already been multiplied by 2 ). This is very different than the true relativisti answer beause the speed is so high. (Note however that if the speed were only 0.10 the relativisti answer would be MeV, and the nonrelativisti answer would be MeV. So they would be almost idential even at 10% of the speed of light). 6. ADDITION OF VELOCITIES Finally, we disuss the relativisti addition of veloities. Suppose that you are riding in a roket ship going a ertain speed v with respet to a stationary observer. You throw a ball out that has a speed of u relative to you. What speed will the observer think the ball is going. You would first think that the stationary observer sees the ball going u = v + u, but that is not orret. Again Einstein showed that the orret relationship should be u = v + u vu We see that when v and u are both less than or equal to then the sum of these two veloities, u, an never be greater than the speed of light itself. When solving problems, remember that u is the veloity as viewed in a frame of referene that is at rest relative to the seond objet, and v is the veloity of that frame of referene relative to one in whih both objets are moving. (In problem 46 u and v are given, but not u ). PROBLEM: On a spaeship traveling at 0.80 a bullet is shot whih travels at 0.70 relative to the spaeship. What speed does an observer on earth measure the bullet to be traveling? 6

7 PROBLEM: On a spaeship traveling at 0.80 headlights are turned on whih have a veloity of relative to the spaeship. What speed does an observer on earth measure the light from the headlights to be traveling? 7