364
6 Special Theory of Relativity
(λ
− λ)/λ becomes (1 − γ )/γ
But γ = 1 + T /Mc
2
= 1 + 120/12 × 10 = 1.01
Hence (λ
− λ)/λ = −0.01/1.01 = −0.0099
6.109 The observed frequency v due to Doppler effect is given by
ν = γ ν
(1 + β cos θ
)
Where ν
is the natural frequency
When the star is moving toward the observer θ
= 0
β = v/c = (300 km/s)/3 × 10
5 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
tan θ = sin θ
∗
/γ c (cos θ
∗
+ β c /β
∗ ).
(1)
For one of the photons, in the rest system of π
◦
.θ
∗
= 90
◦
.β
∗
= 1. From
the given value β c = 0.8 we find γ c = 1.6666. Inserting these values in (1)
tan θ = 0.75 or θ = 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.
tan θ = sin θ
∗
/γ c (cos θ
∗
+ β c /β
∗ )
Use θ = 45
◦ , θ
∗
= 60
◦ , β
∗
= 1, γ c = 1/(1 − β c
2 )
1/2 in the above formula,
and simplify to obtain a quadratic equation in β c . On solving this equation
we find the velocity of the object
v = β c c = 0.535 c
6.112 (a) The y-component of the rod is unchanged that is L y = L y
or L sin θ = L
sin θ
(1)
Also L x = L cos θ
(2)
L
x = L
cos θ
(3)
Eliminating L and L
,
(4)
L x . tan θ = L x
. tan θ
(5)
(b) L x
= L x /γ
where γ is the Lorentz factor. Using (4) in (5)