Chapter 9 Relativeity Chapter Outline 9-1 The Postulate t of Speial Relativity it 9- The Relativity of Time and Time Dilation 9-3 The Relativity of Length and Length Contration 9-4 The Relativisti Addition of Veloities 9-5 Relativisti Momentum and Mass 9-6 Relativisti Energy and E= m
9-5 Relativisti Momentum and Mass When a objet approahes the light speed, the lassial momentum express p = mv, is not valid. For examples, if a large mass with a speed v ollides with a small mass at rest, the small mass an get a speed v; This is not valid if the large mass has a speed v larger than 0.5, sine the speed of the small mass annot be greater than. It an be shown that the orret relativisti momentum for the magnitude : Relativisti Momentum p = mv v SI unit: kg.m/s 9 5
The differene of the relativisti and the lassial momentum Figure 9-13 Relativisti Momentum
Exerise 9-3 Find (a) the lassial and (b) the relativisti momentum of a.4 kg mass moving with a speed of 0.81.
Solution (a) For lassial momentum, p=mv = (.4kg)(0.81x3.00x10 00 10 8 m/s) = 5.8 x 10 8 kg.m/s (b) For relativisti momentum, p 8 mv (.4kg)(0.81 3.00 10 m / s) 8 = = = 9.99 10 kg. m / s v (0.81) The relativisti momentum is always larger than that of the lassial!
Example 9-5 The Missing Mass A satellite, initially iti at rest in spae, explodes into two piees. One piee has a mass of 150kg and moves away from the explosion with a speed of 0.76. The other piee moves away in a opposite diretion with a speed of 0.88. Find the mass of the seond piee of the satellite.
Solution). The magnitude of the momentum for the piee 1 with m 1 =150kg: 1). The magnitude of the momentum for the piee 1 with m 1 =150kg: 8 m1v (150kg)(0.76 3.00 10 m / s) p1 = = = 5.3 10 v1 (0.76) 10 kg. m / s ). The magnitude of the momentum of the piee : p = m v 8 ( m)(0.88 3.00 10 m / s) v = (0.88)
3) )p = p 1 : 8 ( m )(0.88 3.00 10 m / s) 10 = 5.3 10 (0.88) kg. m / s So, m =95kg
The mass inreasing In Equation 9-5, we have m0v m0 p = = ( ) v = v v mv The mass inreasing with speed v as m0 m = 9 6 v Note: 1) When v = 0, m = m 0 ; ) When v approahes, m approahes infinite.
9-6 Relativisti Energy and E= m Sine mass inreases at high speed, when work is done on an objet: 1) part of the work is used to inrease the speed ; ) and part is used to inrease its mass! Considering an objet with mass m 0 at rest. When an objet moves with a speed v, its total t energy is given as: Relativisti Energy E m = m 0 = 9 v 7 SI unit: J Need infinite energy to ahieve the speed of light!
Instead, the energy of an objet at rest, the rest energy E 0 is: Rest Energy with rest mass m 0 E = m0 9 9 SI unit: J This is why material an be onverted into nulear energy!
Exerise 9-4 Find the rest energy of a 0.1-kg apple. Solution: E 0 = m 0 = (0.1kg)(3.00x10 8 m/s) = 1.1 x 10 16 J It ould supply the energy needs of the entire United State for about one hour!
Example 9-6 The Energy of the Sun Energy is radiated by the Sun at the rate of about 3.9x10 6 W. Find the orresponding derease in the Sun s mass for every seond that it radiates.
Solution: 1) Calulate the energy (power) radiated by the Sun in 1.00 s: p= 3.9x10 6 W = 3.93x10 6 J/s. So, ΔE = p Δt = (3.9x10 6 J/s)(1.00s) = 3.9x10 6 J ) Calulate the rest mass: Δm= ΔE E/( ( ) =(3.9x10 6 J)/(3.00x10 18 m/s) = 436 4.36 x10 9 kg This is only a small amount of the total mass of the Sun! The mass loss of the Sun in 1,500 years is only 10-10 of the Sun.
Relativisti Kineti Energy When work is done on a rest objet, its speed inreases, and thus total energy inreases (Equation 9-7). The inrease in the energy beause of the speed, ompared with the rest energy, is all (relativisti) Kineti energy K: Relativisti Kineti Energy m0 E = = m0 + v m0 K = m0 9 9 v SI unit: J K
Compared with the lassi Kineti energy 1 m v 0 Figure 9-16 Relativisti and Classial Kineti Energies
Example 9-7 Relativisti Kineti Energy An observer wathing a high-speed spaeship passing by noties that a lok on board runs slow by a fator of 1.50. If the rest mass of the lok is 0.30kg, what is its kineti energy. Example 9-7 Relativisti Kineti Energy
Solution 1) Using time dilation to alulate the speed v: Δ t i. e. = 1 Δ t Δ Δ 0, v t t So, v = 0.745, that is v/ = 0.745 0 = 1. 5 = 1 1 ( v ) ) Calulate kineti energy K: m (0.30kg)(3.00 10 m/ s) K = v (0.745) 8 0 8 m ) 0 = (0.30km)(3.00 10 m/ s
K (0.30kg)(3.00 10 = (0.745) =1.44 10 16 J 8 m/ s) (0.30km)(3.00 10 8 m/ s) Comparison, the lassial kineti energy is 7.99x10 15 J, always less than that of the relativisti!
A little more on Relativity Theories: 1) Speial Relativity: Disussed until now, There is a speed differene in the two referene frames/systems: no aeleration ) General Relativity: No disussion, There is a aeleration differene in the two referene frames/systems.
Homework of Chapter 9 Due next Wednesday (De 13) Problems (Beginning from page 976):, 16, 4, 8, 38, 4, 46, 55, 56