The length of a rod, of length 5m in a frame of reference which is moving with 0.6 c velocity in a direction making 0 30 angle with the rod is nearly *
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PHYSICS
Find the proper length of a rod if in the laboratory frame of reference its velocity is v=
2
c
, the length l=1.00m, and the angle between the rod and its direction of motion is θ=45
∘
.
November 22, 2019
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Rajarshi Oshinsoasis
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ANSWER
In the rest frame, the coordinates of the ends of the rod in terms of proper length l
0
A:(0,0,0)B:(l
0
cosθ
0
,l
0
sinθ
0
,0)
at time t. In the laboratory frame the coordinates at time t
′
are
A:(vt
′
,0,0),B:(l
0
cosθ
0
1−β
2
+vt
′
,l
0
sinθ
0
,0); where β=
c
v
Therefore we can write,
lcosθ
0
=l
0
cosθ
0
1−β
2
and lsinθ
0
=l
0
sinθ
0
Hence l
0
2
=(l
2
)(
1−β
2
cos
2
θ+(1−β
2
)sin
2
θ
)
or, =
1−β
2
1−β
2
sin
2
θ
∵θ=45
0
and v=
2
c
l
0
=1.08m