College Physics B - PHY2054C. Special Relativity 11/10/2014. My Office Hours: Tuesday 10:00 AM - Noon 206 Keen Building.

Transcription

1 College - PHY2054C 11/10/2014 My Office Hours: Tuesday 10:00 AM - Noon 206 Keen Building

2 Outline 1 2 3

3 1 The speed of light is the maximum possible speed, and it is always measured to have the same value by all observers.

4 Speed of Light

5 1 The speed of light is the maximum possible speed, and it is always measured to have the same value by all observers. 2 There is no absolute frame of reference, and no absolute state of rest.

6 Reference Frames A reference frame can be thought of as a set of coordinate axes. Inertial reference frames move with a constant velocity. The principle of Galilean relativity is the idea that the laws of motion should be the same in all inertial frames. For example, adding or subtracting a constant velocity does not change the acceleration of an object and if Newton s Second Law ( F = m a) is obeyed in one inertial frame, it is obeyed in all inertial frames.

7 The term relativity arises when a situation is described from two different points of view. When the railroad car moves with a constant velocity, Ted and Alice see different motions of the ball.

8 Question 1 Ted travels in a railroad car at constant velocity while his motion is watched by Alice, who is at rest on the ground. Ted s speed v is much less than the speed of light. Ted releases a ball from his hand and observes that in his reference frame the ball falls directly downward. Hence, according to Ted, the component of the ball s velocity along the horizontal direction is zero. According to Alice, what is the ball s velocity along x just after the ball is released? A zero B v in +x direction C v in x direction D v/2 in +x direction

9 Question 1 Ted travels in a railroad car at constant velocity while his motion is watched by Alice, who is at rest on the ground. Ted s speed v is much less than the speed of light. Ted releases a ball from his hand and observes that in his reference frame the ball falls directly downward. Hence, according to Ted, the component of the ball s velocity along the horizontal direction is zero. According to Alice, what is the ball s velocity along x just after the ball is released? A zero B v in +x direction C v in x direction D v/2 in +x direction

10 Question 2 Ted travels in a railroad car at constant velocity while his motion is watched by Alice, who is at rest on the ground. Ted s speed v is much less than the speed of light. Ted releases a ball from his hand and observes that in his reference frame the ball falls directly downward. Hence, according to Ted, the component of the ball s velocity along the horizontal direction is zero. According to Alice, what is the ball s velocity along x just before the ball lands at Ted s feet? A zero B v in +x direction C v in x direction D v/2 in +x direction

11 Question 2 Ted travels in a railroad car at constant velocity while his motion is watched by Alice, who is at rest on the ground. Ted s speed v is much less than the speed of light. Ted releases a ball from his hand and observes that in his reference frame the ball falls directly downward. Hence, according to Ted, the component of the ball s velocity along the horizontal direction is zero. According to Alice, what is the ball s velocity along x just before the ball lands at Ted s feet? A zero B v in +x direction C v in x direction D v/2 in +x direction

12 Question 3 Ted travels in a railroad car at constant velocity while his motion is watched by Alice, who is at rest on the ground. Ted s speed v is much less than the speed of light. Ted releases a ball from his hand and observes that in his reference frame the ball falls directly downward. Hence, according to Ted, the component of the ball s velocity along the horizontal direction is zero. According to Alice, what is the acceleration of the ball along x? A zero B g in +y direction C g in y direction D g/2 in y direction

13 Question 3 Ted travels in a railroad car at constant velocity while his motion is watched by Alice, who is at rest on the ground. Ted s speed v is much less than the speed of light. Ted releases a ball from his hand and observes that in his reference frame the ball falls directly downward. Hence, according to Ted, the component of the ball s velocity along the horizontal direction is zero. According to Alice, what is the acceleration of the ball along x? A zero B g in +y direction C g in y direction D g/2 in y direction

14 Question 4 Ted travels in a railroad car at constant velocity while his motion is watched by Alice, who is at rest on the ground. Ted s speed v is much less than the speed of light. Ted releases a ball from his hand and observes that in his reference frame the ball falls directly downward. Hence, according to Ted, the component of the ball s velocity along the horizontal direction is zero. According to Ted, what is the force F x on the ball (m = 0.6) kg along x? A zero B mg in +y direction C mg in y direction D mg/2 in y direction

15 Question 4 Ted travels in a railroad car at constant velocity while his motion is watched by Alice, who is at rest on the ground. Ted s speed v is much less than the speed of light. Ted releases a ball from his hand and observes that in his reference frame the ball falls directly downward. Hence, according to Ted, the component of the ball s velocity along the horizontal direction is zero. According to Ted, what is the force F x on the ball (m = 0.6) kg along x? A zero B mg in +y direction C mg in y direction D mg/2 in y direction

16 Question 5 Ted travels in a railroad car at constant velocity while his motion is watched by Alice, who is at rest on the ground. Ted s speed v is much less than the speed of light. Ted releases a ball from his hand and observes that in his reference frame the ball falls directly downward. Hence, according to Ted, the component of the ball s velocity along the horizontal direction is zero. Do Ted and Alice agree on the value of F x? A yes B no

17 Question 5 Ted travels in a railroad car at constant velocity while his motion is watched by Alice, who is at rest on the ground. Ted s speed v is much less than the speed of light. Ted releases a ball from his hand and observes that in his reference frame the ball falls directly downward. Hence, according to Ted, the component of the ball s velocity along the horizontal direction is zero. Do Ted and Alice agree on the value of F x? A yes B no

18 Ted observes the ball s motion purely along the vertical. Alice sees projectile motion in both the x- and y-directions. Both agree that a y = g (due to gravity) and a x = 0. Newton s Second Law is obeyed.

19 Galilean and Light According to Maxwell s equations, the speed of light, c, has a constant value: He also showed that the speed of light is independent of the motion of both the source and the observer.

20 Galilean and Light According to Maxwell s equations, the speed of light, c, has a constant value: He also showed that the speed of light is independent of the motion of both the source and the observer. 1. Newton s mechanics predict that the speed of the light wave relative to Alice should be c + v. 2. According to Maxwell s theory, Ted and Alice should both observe the light wave to move with speed c.

21 Michelson-Morley Experiment 1887: Michelson and Morley attempted to determine Earth s motion relative to the absolute space through which light supposedly moved by measuring the speed of light at different times of the day and on different days of the year.

22 Michelson-Morley Experiment 1887: Michelson and Morley attempted to determine Earth s motion relative to the absolute space through which light supposedly moved by measuring the speed of light at different times of the day and on different days of the year. Far from measuring the properties of absolute space, the experiment demolished the entire concept.

23 Galilean and Light Galilean and electromagnetism do predict different results for observers in different inertial frames: Experiments showed that Maxwell s theory was correct. The speed of light in the vacuum is always c. Galilean relativity for how the speed of light depends on the motion of the source is wrong. Einstein developed theory of relativity:. Two Postulates 1 All laws of physics are the same in all inertial reference frames. 2 The speed of light in the vacuum is a constant.

24 Inertial Reference Frames The modern definition of an inertial reference is one in which Newton s First Law holds: If a particle moves with a constant velocity, then the reference frame is inertial. Earth s acceleration is small enough that it can be ignored (can be considered an inertial system).

25 Outline 1 2 3

26 Light Clock The two postulates lead to a surprising result concerning the nature of time. A light clock keeps time by using a pulse of light that travels back and forth between two mirrors: The time for the clock to tick once is the time needed for one round trip:

27 Moving Light Clock The clock moves with a constant velocity v relative to the ground: From Ted s reference frame, the light pulse travels up and down between the two mirrors: t 0 = 2l/c.

28 Moving Light Clock The clock moves with a constant velocity v relative to the ground: From Ted s reference frame, the light pulse travels up and down between the two mirrors: t 0 = 2l/c. Alice sees the light pulse travel a longer distance, but the speed of light is the same for Alice as for Ted. Because of the longer distance, according to Alice the light will take longer to travel between the mirrors.

29 Moving Light Clock For Alice, the time for one tick of the clock is: t = t 0 1 v2 c 2 The time for Ted is different from the time for Alice. The operation of the clock depends on the relative motion.

30 Special relativity predicts that moving clocks run slow. This effect is called. For typical terrestrial speeds, the difference between t and t 0 is negligible. t 0 is called the proper time: t = t 0 1 v2 c 2 = 1 t 0 (100 mph)2 c 2 t 0 1

31 When the speed v is small compared to c, the factor 1 v 2 /c 2 is very close to 1:

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33 Outline 1 2 3

34 Lorentz γ = 1 1 v 2 /c 2 : L 0 = v t L = v t 0

35 Lorentz When measuring the length of the moving meterstick, you do so by noting the positions of the two ends at the same time, according to your clock. However, those two events the two measurements you make do not occur at the same time as seen by the moving observer. In relativity, time is relative, and simultaneity (the idea that two events happen at the same time ) is no longer a well-defined concept.

36 1 The speed of light is the maximum possible speed, and it is always measured to have the same value by all observers. 2 There is no absolute frame of reference, and no absolute state of rest. 3 Space and time are not independent, but are unified as spacetime.

37 Relativistic Addition of Velocities: v = v 1 + v v 1 v 2 c 2 1 When two velocities are much less than the speed of light, the relativistic addition of velocities gives nearly the same result as the Newtonian equation. Okay for speeds less than 10% of the speed of light! 2 Experiments with particles moving at very high speeds show that the relativistic result is correct.