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Freely expanding rotating particle ring
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Past exam of the mathematics course of the University of Cambridge
/
2024
/
ia
/
Paper 4
/
11C
/
b
/
ii
/
Solution
Created
2026-09-24
Updated
2026-09-25
View more
Each freely moving particle has
position
, relative to its own initial radial and tangential unit
vectors
,
x
i
(
t
)
=
R
,
e
r
,
i
+
v
t
,
e
θ
,
i
.
(1)
Thus all particles lie on
a
circle
of
radius
R
(
t
)
=
R
2
+
v
2
t
2
.
(2)
Their
angular momenta
are constant and
sum
to
L
=
m
R
v
,
z
^
.
(3)
Since
I
(
t
)
=
m
R
(
t
)
2
, the angular
velocity
is
θ
˙
=
I
(
t
)
L
=
R
2
+
v
2
t
2
R
v
.
(4)
With
θ
(
0
)
=
0
,
θ
(
t
)
=
arctan
R
v
t
.
(5)
This is the
geometry
of
a
freely expanding rotating particle ring
.
Total
articles
:
1