Chapter 26. Special Relativity

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1 Chapter 26 Special Relativity

2 The Postulates of Special Relativity THE POSTULATES OF SPECIAL RELATIVITY 1. The Relativity Postulate. The laws of physics are the same in every inertial reference frame. 2. 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.

3 Some consequences of Einstein s two postulates of Special Relativity Time dilation Δt γδt 0 Δt Δt 0 Length contraction L L 0 γ L L 0 where, Δt 0 proper time, ΔL 0 proper length, and γ 1 1 v2 1

4 Some consequences of Einstein s two postulates of Special Relativity -- continued For a moving object: Mass Increase m γm 0 m m 0 m relativistic mass, m 0 rest mass Relativistic momentum p mv γm 0 v Total relativistic energy E m γm 0

5 Relativistic Momentum Comparison between relativistic and nonrelativistic momentum Relativistic momentum Nonrelativistic momentum p m 0v 1 v2 p classical m 0 v

6 The Equivalence of Mass and Energy RELATIVISTIC ENERGY OF AN OBJECT Total energy of an object (with PE 0) E γm 0 m 0 1 v 2 Rest energy of an object (v 0) E o m 0 à Intrinsic energy of an object with rest mass m 0 Kinetic Energy KE E E o m 0 ( γ 1)

7 Kinetic energy in the classical limit of v à 0 : Binomial theorem " $ KE m 0 ( γ 1) m 0 $ 1 $ $ 1 v2 # ( ) % ' 1' ' ' & ( 1+ x) α 1+αx + α α 1 x α x 1 " % 1 1 v2 2 " 1 %" % $ ' 1+ $ ' $ v2 '+ 1+ v2 for v 0 1 v2 # & # 2 &# & 2 " KE m 0 $ 1+ v2 # 2c 1 % ' 1 2 & 2 m 0 v2

8 Useful relationship between E and p : E m p mv m γm 0 γ p 2 m 0 v2 1 v2 1 v2 p 2 p 2 v 2 c m v 2 ( ) p 2 c m p 2 2 m p 2 E 2 m 2 0c 4 1 v2 # % $ m 0 2 c 4 m 0 2 m p 2 m 0 2 m p v2 v2 p 2 m p 2 p 2 + m 2 & 0 c 4 E p 2 + m 2 0 c 4 ( '

9 How much energy does it take to accelerate a mass m 0 to c? v c E m γm 0 m 0 c2 1 c2 m 0 c2 1 1 m 0 c2 0 Therefore, it would take an infinite amount of energy to accelerate any non-zero mass to c à c is the absolute speed limit of the Universe!

10 Van de Graaff generator used as a particle accelerator. An electron is accelerated vertically starting from rest across a +100,000 V potential difference produced by a Van de Graaff. Find the final electron a) kinetic energy, b) rest mass energy, c) total relativistic energy, d) relativistic mass, e) momentum, and f) speed. Compare the result in part f) with the classical calculation. a) ΔKE KE 0 ΔPE qδv ( )( 10 5 ) J 100 KeV b) E 0 m 0 ( )( ) J 512 KeV V B V c) E KE + E J 612 KeV d) E m m E c ( ) kg V A 0 V electron q C, v A 0 m s m electron kg

11 e) E 2 p 2 + m 0 2 c 4 p E 2 m 0 2 c 4 c kg m s ( ) 2 ( ) 2 ( ) f ) p mv v p m m s c From classical mechanics: KE 1 2 m 0v 2 v 2KE m ( ) m s Classical mechanics gives a speed that is about 14% too high in this case.

12 The Equivalence of Mass and Energy Example: The Sun is Losing Mass The sun radiates electromagnetic energy at a rate of 3.92 x 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 (m sun 1.99 x kg).

The Equivalence of Mass and Energy Δm 26 (.92 10 J s)( 1.0 s) ΔEo 4.

13 The Equivalence of Mass and Energy Δm 26 ( J s)( 1.0 s) ΔEo c ( m s) kg 75 years Δm m sun 9 7 ( kg s)( s) kg

14 The Equivalence of Mass and Energy Conceptual Example: 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? PE 1 2 kx2

15 Example. A 8 TeV proton from the Large Hadron Collider. The LHC can accelerate protons to 8 TeV kinetic energy. Find γ for an accelerated proton (m proton 1.67 x kg). What is its speed? E m γm 0 E 0 m 0 ( )( ) J 0.94 GeV γ E m 0 c KE + E 0 2 m GeV GeV 0.94 GeV 8500 Proton speed at 8 TeV kinetic energy: γ 1 1 v2 v c 1 1 γ 2 c c Very close to c!

16 Example. Energy needed to accelerate an object to a speed near c. If kinetic energy is supplied to an object equal to its rest mass, what speed will the object have? E KE + E 0 m γm 0 If KE E 0 E 2E 0 γm 0 1 γ m 0 2E 0 m 0 2m # v2 1 2 % γ $ & ( 1 ' 4 γ 2 v 3 2 c c Time slowed to ½ in object s rest frame Not far from c How much energy must be supplied to accelerate a 5 metric ton spacecraft to this speed? E 0 m 0 ( 5000) ( ) J About 1 millionth of the E&M output of the Sun in 1 second (maybe if you had a large enough solar sail over a 1 year period )

Chapter 26 Special Theory of Relativity

Chapter 26 Special Theory of Relativity Classical Physics: At the end of the 19 th century, classical physics was well established. It seems that the natural world was very well explained. Newtonian mechanics