Time dilation

This ppt explains about time dilation, variation of mass with velocity, mass-energy equivalence

1. TIME DILATION Dr.R.Hepzi Pramila Devamani , Assistant Professor of Physics, V.V.Vanniaperumal College for Women, Virudhunagar

2. TIME DILATION Derivation Imagine a gun placed at the position (x' y z) in S' (Fig. 5.7). Suppose it fires two shots at times t 1 ’ and t 2 ’ measured with respect to S'. In S' the clock is at rest relative to the observer. The time interval measured by a clock at rest relative to the observer is called the proper time interval . Hence , to = t 2 ’ –t 1 ’ is the time interval between the two shots for the observer in S '. Since the gun is fixed in S', it has a velocity v with respect S in the direction of the positive X-axis. Let t = t 2 – t 1 represent the time interval between the two shots as measured by an observer in S.

3. TIME DILATION

4. TIME DILATION From inverse Lorentz transformations we have

5. TIME DILATION Thus, the time interval, between two events occurring at a given point in the moving frame S' appears to be longer to the observer in the stationary frame S. Thus a stationary clock measures a longer time interval between events occurring in a moving frame of reference than does a clock in the moving frame. This effect is called time dilation.

6. Explanation Time interval, like a length interval, is not absolute. It is also affected by relative motion . The time-interval between two ticks as judged from moving frame S ' is t o. Suppose another observer measures the time interval between same two ticks as t, from a stationary frame S, relative to which the clock is moving with a speed v. Then, t =  

7. Explanation This equation shows that to the observer in frame S , the time appears to be increased by a factor Thus a stationary clock measures a longer time-interval between events occurring in a moving frame of reference than does a clock in the moving frame.  

8. Explanation In other words, a moving clock appears to be slowed down to a stationary observer . This effect is called time dilation. If v = c, t = , It means that a clock moving with the speed of light will appear to be completely stopped to a stationary observer .  

9. The Twin Paradox Consider two exactly identical twin brothers. Let one of the twins go to a long space journey at a high speed in a rocket and the other stay behind on the earth. The clock in the moving rocket will appear to go slower than the clock on the surface of earth. Therefore, when he returns back to the earth, he will find himself younger than the twin who stayed behind on the earth!

10.   Variation of Mass with Velocity Derivation Consider two systems S and S' S' is moving with a constant velocity V relative to the system S, in the positive X-direction (F1g. 5.8). Suppose that in the system S', two exactly similar elastic balls A and B approach each other at equal speeds (i.e., u and -u). Let the mass of each ball be m in S’. They collide with each other and after collision coalesce into one body. According to the law of conservation of momentum. Momentum of ball A+ momentum of ball B= momentum of coalesced mass. mu + (-mu)= momentum of coalesced mass= 0. Thus the coalesced mass must be at rest in S’ system.

11. Variation of Mass with Velocity

12. Variation of Mass with Velocity Let us now consider the collision with reference to the system S . Let u 1 and u 2 be the velocities of the balls relative to S. Then , After Collision, velocity of the coalesced mass is v relative to the system

13. Variation of Mass with Velocity Let mass of the ball A travelling with velocity u 1 be m 1 and that of B with velocity u 2 be m 2 in the system S . Total momentum of the balls is conserved .

14. Variation of Mass with Velocity

15. Variation of Mass with Velocity

16. Variation of Mass with Velocity

17. Explanation In Newtonian mechanics, the mass of a body is taken constant and independent of velocity. But according to special theory of relativity , the mass of a body changes with its velocity. That is, the mass of a body in motion is different from the mass of the body at rest. The mass of a body varies with its velocity according to the relation m = Here , m is the "rest mass" of the body, c is the velocity of light and v is the velocity of the body.  

18. Explanation Following conclusions are drawn from mass variation formula As the velocity v of the particle relative to the observer increases, the mass of the particle increases. As v This means that no material particle can have a velocity equal to, or greater than, the velocity of light . When v << c, then v 2 / c 2 can be neglected as compared to 1 and so m m 0. This means that at ordinary velocities the difference between m and m is so small, that it can not be detected. The increase in mass has been directly verified in various electron diffraction experiments and also in the operation of particle accelerators  

19. Mass Energy Equivalence Force is defined as rate of change of momentum F = (mv) (1) According to the theory of relativity, both mass and velocity are variable. F = (mv ) = m dx + v dx (2)  

20. Mass Energy Equivalence Let the force F displace the body through a distance dx. Increase in the kinetic energy ( dEk ) of the body is equal to the work done (F dx). Hence, dEk = F dx = m dx + v dx Or dEk = mv dv + v 2 dm ( 3 ) According to the law of variation of mass with velocity , m = (4)  

21. Mass Energy Equivalence squaring both sides, m 2 = m 2 / (1-v 2 /c 2 ) m 2 c 2 = m 2 c 2 + m 2 v 2 Differentiating, c 2 2m dm = m 2 2vdv + v 2 2mdm c 2 dm = mvdv + v 2 dm (5) From Eqs . (3) and (5), dEk = c 2 dm (6) Thus, a change in K.E., dEk is directly proportional to a change in mass dm.

22. Mass Energy Equivalence When a body is at rest, its velocity is zero, (K.E = 0) and m=m , when its velocity is v, its mass becomes m.

23. Mass Energy Equivalence This is the relativistic formula for KE. When the body is at rest, the internal energy stored in the body is m c 2. m c 2 called the rest mass energy. The total energy (E) of the body is the sum of KE( Ek ) and rest mass energy ( m c 2 ). E = Ek + m c 2 = m c 2 - m c 2 + m c 2 = m c 2 E = m c 2 . This is Einstein's mass-energy relation.

24. Mass Energy Equivalence Explanation According to the theory of relativity, if the mass m of a particle is completely converted into energy , then E = m c 2 Here , c is the velocity of light. This is the famous Einstein's mass-energy relation

25. Mass Energy Equivalence It states that an energy m c 2 associated with a mass m. Conversely , a mass E / c 2 associated with energy. This relation states a universal equivalence between mass and energy. It means that mass may appear as energy as mass. The relationship (E = m c 2 ) between energy and mass forms the basis of understanding nuclear reactions such as fission and fusion. These reaction take place in nuclear bombs and reactors. When uranium nucleus is split up, the decrease in its total rest mass appears in the form of an equivalent amount of K.E, of its fragments .