PHY3063 Spring 2006
R. D. Field
Exam 1 Solutions
Page 1 of 7
January 31, 2006
PHY 3063 Exam 1 Solutions
Problem 1 (25 points):
A rod with is at rest in the
Oframe
(
parallel to the
xaxis
) with the left end of the rod (
L
) at
x = 0
and
right end (
R
) at
x = R
and a cart with rest length
L
0
= 300 m
is at rest in the
O'frame
(
parallel to the
x'axis
) with the right end of the cart (
r
) at
x' = 0
and left end (l) at
x' = L
0
. as shown in the
figure
.
The
O'frame
is moving to the right along the
positive xaxis of the
Oframe
with velocity
V =
0.8c
and the origins of the two frames coincide at
t
= t' = 0
. (Note that c = 3×10
8
m/s.)
Consider the
following four events:
Event
Description
A
Right end of cart crosses left end of rod
(Lr)
B
Right end of cart crosses right end of rod
(Rr)
C
Left end of cart crosses left end of rod
(Ll)
D
Left end of cart crosses right end of rod
(Rl)
In the
Oframe
event
B
and event
C
occur
simultaneously
(
i.e.
0
=
−
=
∆
B
C
t
t
t
).
Part A (10 points):
Plot the four events
A
,
B
,
C
, and
D
on the spacetime plots below and label
the (ct, x) coordinates of each event in the
Oframe
and the (ct',x') coordinates of each event in
the
O'frame
.
(Express the coordinates in terms of c,
β
,
γ
, and L
0
).
.
Oframe
x
ct
Path of r
Path of l
A = (0,0)
(L
0
/
βγ
,0) = C
B = (L
0
/
βγ
,L
0
/
γ
)
D = (2L
0
/
βγ
,L
0
/
γ
)
θ
O'frame
x'
ct'
Path of L
Path of R
θ
(0,0) = A
B = (L
0
/
βγ
2
,0)
(L
0
/
β
, L
0
) = C
D = (L
0
(2
β
2
)/
β
,L
0
)
Solution:
Oframe:
The key to plotting the events is to first plot the paths of the right (r) and left (l) end of
the cart in the Oframe.
Both r and l follow straight (parallel) lines with
β
θ
/
1
tan
=
.
Events A
and C lie at x = 0 and events B and D lie at x = R. Also,
/
R
ct
B
=
and
)
/(
/
0
βγ
L
L
ct
C
=
=
,
where I used
γ
/
0
L
L
=
. Thus,
/
0
L
R
=
and
)
/(
2
0
L
ct
D
=
.
O'frame:
We can use the Lorentz transformations to compute the coordinates of the four events
in the O'frame as follows:
y
x
z
y'
z'
x'
V
O
O'
Cart at rest in
the O'frame.
Rod at rest in the
Oframe.
L
R
l
r
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R. D. Field
Exam 1 Solutions
Page 2 of 7
January 31, 2006
0
)
(
'
0
)
(
'
=
−
=
=
−
=
A
A
A
A
A
A
ct
x
x
x
ct
ct
β
γ
0
0
0
0
)
(
'
)
(
'
L
L
ct
x
x
L
L
x
ct
ct
C
C
C
C
C
C
−
=
−
=
−
=
=
=
−
=
βγ
0
)
(
'
)
1
(
)
(
'
0
0
2
0
2
0
0
0
=
−
=
−
=
=
−
=
−
=
−
=
L
L
ct
x
x
L
L
L
L
x
ct
ct
B
B
B
B
B
B
0
0
0
2
0
0
0
0
0
2
)
(
'
)
2
(
2
2
)
(
'
L
L
L
ct
x
x
L
L
L
L
L
x
ct
ct
D
D
D
D
D
D
−
=
−
=
−
=
−
=
−
=
−
=
−
=
Both R and L follow straight (parallel) lines with
θ
/
1
tan
=
.
Events A and B lie at x' = 0 and
events C and D lie at x' = L
0
. Also,
/
'
0
L
ct
C
=
,
)
/(
'
2
0
L
ct
B
=
, and
/
)
2
(
)
/(
/
'
2
0
2
0
0
−
=
−
=
L
L
L
ct
D
.
Part B (5 points):
How long (
in microseconds
) does it take for the cart to cross over the rod
according to an observer in the
Oframe
(
i.e.
what is
A
D
t
t
t
−
=
∆
)?
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 Spring '07
 Field
 Physics, Momentum, GeV, R. D., R. D. Field

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