Florida State University
Department of Physics
PHY 5246
Assignment # 1 (Due Wednesday, September 9th, 2009)
(1) (10 points) A massless string is placed over a masssless pulley, and each end is wound
around and fastened to a vertical hoop. The hoops have masses
M
1
and
M
2
, and radii
R
1
and
R
2
. The apparatus is placed in a uniform gravitational Feld
g
and released with each end of
the string aligned along the Feld.
(a) Show that the tension in the string is
τ
=
gM
1
M
2
/
(
M
1
+
M
2
).
(b) Show that the acceleration of the center of mass of hoop 1 is
a
1
=
M
1
(
M
1
−
M
2
)

1
a
point
1
,
where
a
point
1
is the acceleration when both hoops are replaced with point masses.
(2) (10 points) (a) Starting from Newton’s second law
m
¨
r
=
F
for a central force
F
=
f
(
r
)ˆ
r
,
show that the equations of motion in twodimensional polar coordinates become
m
(¨
r
−
r
˙
φ
2
) =
f
(
r
)
and
m
(2˙
r
˙
φ
+
r
¨
φ
) = 0
.
(b) Integrate these equations explicitly; hence relate them to the treatment of centralforce
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 Fall '09
 ROBERTS
 mechanics, Mass, Florida State University, circular orbit, Celestial mechanics, emin, stable circular orbit

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