Special Relativity with solved problems (1)

For university students

1. Chapter 28 Special Relativity

2. 28.1 Events and Inertial Reference Frames An event is a physical “happening” that occurs at a certain place and time. To record the event, each observer uses a reference frame that consists of a coordinate system and a clock. Each observer is at rest relative to her own reference frame. An inertial reference frame is one in which Newton’s law of inertia is valid.

3. 28.1 Events and Inertial Reference Frames In this example, the event is the space shuttle lift off.

4. 28.2 The Postulates of Special Relativity THE POSTULATES OF SPECIAL RELATIVITY The Relativity Postulate. The laws of physics are the same in every inertial reference frame. The Speed of Light Postulate. The speed of light in a vacuum, measured in any inertial reference frame, always has the same value of c , no matter how fast the source of light and the observer are moving relative to one another.

5. 28.2 The Postulates of Special Relativity

6. 28.3 The Relativity of Time: Time Dilation TIME DILATION A light clock

7. 28.3 The Relativity of Time: Time Dilation Suppose two identical clocks are built. One is kept on earth, and the other is placed aboard a spacecraft that travels at a constant velocity relative to the earth The astronaut is at rest with respect to the clock on the spacecraft and, therefore, sees the light pulse move along the up/down path shown in top diagram. According to the astronaut, the time interval required for the light to follow this path is the distance 2 D divided by the speed of light c ;   An earth-based observer , however, does not measure as the time interval between these two events.   Since the spacecraft is moving , the earth-based observer sees the light pulse follow the diagonal path shown in red in part b of the drawing . This path is longer than the up/down path seen by the astronaut. But light travels at the same speed c for both observers , in accord with the speed-of-light postulate

8. An observer on the earth sees the light pulse travel a greater distance between ticks. The time interval that the earth-based observer measures in Figure b can be determined as follows   While the light pulse travels from the source to the detector , the spacecraft moves a distance to the right, where v is the speed of the spacecraft relative to the earth.   From the drawing it can be seen that the light pulse travels a total diagonal distance of 2 s during the time interval   Applying the Pythagorean theorem, we find that But the distance 2 s is also equal to the speed of light times the time interval so that  

9. and Therefore solving for we get   But , where is the time interval between successive “ticks” of the spacecraft’s clock as measured by the astronaut. Substituting we get the equation for TIME DILATION given by  

10. 28.3 The Relativity of Time: Time Dilation Time dilation The symbols in this formula are defined as follows:

11. 28.3 The Relativity of Time: Time Dilation PROPER TIME INTERVAL The time interval measured at rest with respect to the clock is called the proper time interval . In general, the proper time interval between events is the time interval measured by an observer who is at rest relative to the events. Proper time interval Check your understanding 1 to 3 page 876

12. 28.3 The Relativity of Time: Time Dilation Example 1 Time Dilation The spacecraft is moving past the earth at a constant speed of 0.92 times the speed of light. The astronaut measures the time interval between ticks of the spacecraft clock to be 1.0 s. What is the time interval that an earth observer measures?

13. 28.4 The Relativity of Length: Length Contraction The shortening of the distance between two points is one example of a phenomenon known as length contraction . Length contraction

14. 28.4 The Relativity of Length: Length Contraction Example 4 The Contraction of a Spacecraft An astronaut, using a meter stick that is at rest relative to a cylindrical spacecraft, measures the length and diameter to be 82 m and 21 m respectively. The spacecraft moves with a constant speed of 0.95 c relative to the earth. What are the dimensions of the spacecraft, as measured by an observer on earth.

15. 28.4 The Relativity of Length: Length Contraction Diameter stays the same.

16. 28.5 Relativistic Momentum Relativistic momentum

17. 28.6 The Equivalence of Mass and Energy THE TOTAL ENERGY OF AN OBJECT Total energy of an object Rest energy of an object When an object is accelerated from rest to a speed v , the object acquires kinetic energy in addition to its rest energy. The total energy E is the sum of the rest energy E and the kinetic energy KE, E = E + KE Kinetic energy of an object

18. The Relation Between Total Energy and Momentum It is possible to derive a useful relation between the total relativistic energy E and the relativistic momentum p . From the equation of relativistic momentum, make the subject of formula Substitute on the equation for energy   Substitute on the equation for energy   Solving the equation for energy we get  

19. Why the Speed of Light in a Vacuum Is the Ultimate Speed One of the important consequences of the theory of special relativity is that objects with mass cannot reach the speed c of light in a vacuum Consider Equation, The equation which gives the kinetic energy of an object moving at a speed v As v approaches the speed of light c, the denominator Approaches zero Hence, the kinetic energy becomes infinitely large However, the work–energy theorem tells us that an infinite amount of work would have to be done to give the object an infinite kinetic energy . Since an infinite amount of work is not available , we are left with the conclusion that objects with mass cannot attain the speed of light c . Thus, c represents the ultimate speed .

20. Problems 1 to 20

21. 28.6 The Equivalence of Mass and Energy Example 8 The Sun is Losing Mass The sun radiates electromagnetic energy at a rate of 3.92x10 26 W. What is the change in the sun’s mass during each second that it is radiating energy? What fraction of the sun’s mass is lost during a human lifetime of 75 years.

22. 28.6 The Equivalence of Mass and Energy

23. The total momentum of the man/woman system is conserved, since no net external force acts on the system. Therefore, the final total momentum p m + p w must equal the initial total momentum, which is zero. As a result, p m  = – p w Solving for v m , we find

24. According to the work-energy theorem , he work that must be done on the electron to accelerate it from rest to a speed of 0.999 c is equal to the kinetic energy of the electron when it is moving at 0.999 c . The mass m of the aspirin is related to its rest energy E Since it requires 1.1  10 8 J to operate the car for twenty miles, we can calculate the number of miles that the car can go on the energy that is equivalent to the mass of one tablet The number N of miles the car can go on one aspirin tablet is

25. According to the work-energy theorem, the change in the kinetic energy equals the work W done to accelerate the electron. Since the electron starts from rest, its initial kinetic energy is zero, so its final kinetic energy KE equals the work done. The relativistic kinetic energy of the electron is Recognizing that the electron’s final kinetic energy KE equals the work W done to accelerate the electron, we have

26. for the speed v , we proceed as follows

27. The total energy for each particle is given by The annihilation energy is twice this value, so .

28. 28.6 The Equivalence of Mass and Energy Conceptual Example 9 When is a Massless Spring Not Massless? The spring is initially unstrained and assumed to be massless. Suppose that the spring is either stretched or compressed. Is the mass of the spring still zero, or has it changed? If the mass has changed, is the mass change greater for stretching or compressing?