PHYS 230
Fall 2007
Assignment 8
1. French, problem 2.10, extended. 10 points. A particle moves along the curve
y
=
Ax
2
such that its
x
position is given by
x
=
Bt
.
(a) Express the vector position of the particle in the form
r
(
t
) =
x
i
+
y
j
.
Since
y
=
Ax
2
=
AB
2
t
2
, we get
r
=
Bt
ˆ
i
+
AB
2
t
2
ˆ
j
.
(b) Calculate the speed
v
(=
ds/dt
) of the particle along this path at an arbitrary
instant.
Differentiate
x
and
y
, separately, with respect to
t
.
We get
v
=
B
ˆ
i
+
2
AB
2
t
ˆ
j
. Therefore, the speed
v
is
v
=
√
B
2
+ 4
A
2
B
4
t
2
.
(c) Show that the angular velocity
ω
= (
dθ/dt
) at an arbitrary instant is
AB/
(1 +
A
2
B
2
t
2
).
The component of the velocity
v
along
ˆ
θ
is just
rdθ/dt
=
r
˙
θ
=

v
x
sin
θ
+
v
y
cos
θ
.
But
sin
θ
=
y/r
and
cos
θ
=
x/r
, so:
r
˙
θ
=

AB
3
t
2
(1

2)
/r
˙
θ
= +
AB
3
t
2
/
(
B
2
t
2
+
A
2
B
4
t
4
) =
AB/
(1 +
A
2
B
2
t
2
)
2. 10 points. A penny is rolling upright in a straight line along a horizontal surface. Its
speed is constant. A red spot is painted at one point on its circumference. Plot a graph
of the position of this spot as a function of time as seen by a stationary observer. Show
that once every revolution the velocity of the red spot is exactly zero.
&%
’$

v
¡
¡
s
θ

ω
The velocity of the red spot has two pieces. They are a constant velocity
v
=
v
ˆ
x
, and
an angular velocity about the centre of the penny. Note, however, that when the rolling
motion is to the right (positive
x
), the rolling motion is clockwise, so that
ω
is negative,
ω
=

v/R
. (This is the condition for rolling . . . )
In the frame moving with velocity
v
ˆ
x
, the position of the red spot is
´
x
=
R
cos
θ
and
´
y
=
R
+
R
sin
θ
, where we choose
θ
=

ωt

π/
2
, so that
x
=
y
= 0
at
t
= 0
.
In the frame of the surface, at time
t
, the centre of the penny is at
(
vt, R
)
. The red
spot is therefore at
(
vt
+
R
cos
θ, R
+
R
sin
θ
)
. When you draw this, you get a socalled
cycloid. The graph has
R
= 1
,
ω
= 2
π
.
1
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0
5
10
15
20
25
0
1
2
3
4
The velocity is just
d
dt
v
. We find this by differentiating each component independently
. . .
v
x
=
v
+
Rω
sin

ωt

π/
2 =
v
(1

sin
ωt
+
π/
2)
v
y
=

Rω
cos

ωt

π/
2 =

v
cos
ωt
+
π/
2
By inspection, whenever
t
=
n
2
π/ω
;
n
= 0
,
1
,
2
. . . , we get both
v
x
and
v
y
= 0
. This
is whenever the red spot is on the ground:
y
= 0
.
3. French, problem 2.16 (part (a) only). Slightly modified, for clarity. 10 points. This
problem is about the relationship between polar and Cartesian coordinates.
Research Project: complete the questionnaire for this question.
The orbital radius of Venus is 0.72 times the radius of Earth’s orbit. Its period is about
0.62 times Earth’s year. Using these data, find how the apparent angular positions of
Venus changes with time as seen from Earth, assuming that the orbits of both planets
lie in the same plane. Present your results as a plot of angular position versus time,
and compare them semiquantitatively with Figure 220 (see next page).
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 Fall '07
 Harris
 Angular Momentum, Kinetic Energy, Kepler's laws of planetary motion

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