1000 Solved Problems in Modern Physics

364 6 Special Theory of Relativity

(λ′−λ)/λbecomes (1−γ)/γ

Butγ= 1 +T/Mc^2 = 1 + 120 / 12 × 10 = 1. 01

Hence (λ′−λ)/λ=− 0. 01 / 1. 01 =− 0. 0099

6.109 The observed frequencyvdue to Doppler effect is given by

ν=γν′(1+βcosθ′)

Whereν′is the natural frequency

When the star is moving toward the observerθ′= 0

β=v/c=(300 km/s)/ 3 × 105 km/s= 10 −^3

γ= 1 /(1−β^2 )^1 /^2 ≈ 1 +(1/2)β^2 = 1 + 5 × 10 −^7

Neglecting small terms,v=(1+ 10 −^3 )ν′

Fractional change in frequency

(ν−ν′)/ν′= 10 −^3

6.110 Use the formula for Lorentz transformation of angles from CMS to LS

tanθ=sinθ∗/γc(cosθ∗+βc/β∗). (1)

For one of the photons, in the rest system ofπ◦.θ∗= 90 ◦.β∗=1. From

the given valueβc= 0 .8 we findγc= 1 .6666. Inserting these values in (1)

tanθ = 0 .75 orθ= 36. 87 ◦in the LS. From symmetry the second photon

will also be emitted at the same angle on the other side of the line of flight and

be coplanar. Hence the angle between the two photons will be 2θ= 73. 75 ◦

6.111 Use the formula for the transformation of angles.

tanθ=sinθ∗/γc(cosθ∗+βc/β∗)

Useθ= 45 ◦,θ∗= 60 ◦,β∗=1,γc= 1 /(1−βc^2 )^1 /^2 in the above formula,

and simplify to obtain a quadratic equation inβc. On solving this equation

we find the velocity of the object

v=βcc= 0. 535 c

6.112 (a) They-component of the rod is unchanged that isLy=Ly′

orLsinθ=L′sinθ′ (1)

AlsoLx=Lcosθ (2)

L′x=L′cosθ′ (3)

EliminatingLandL′, (4)

Lx.tanθ=Lx′.tanθ′ (5)

(b)Lx′=Lx/γ

whereγis the Lorentz factor. Using (4) in (5)