Phys 2101
Homework 4 Solution Fall `06
The numerical answers to the problems are calculated with the values used in the problems found
in the printed textbook; the values for your particular WebAssign assignment will be different.
1. CQ.10-09.
We have three flat disks, with the same size (same inner circle radius
R
c
and the
same outer shell radius R
s
), the same two materials with densities
1
ρ
and
2
ρ
with
1
ρ
>
2
ρ
, with
the area of the inner circle equal to the area of the outer shell,
A
c
= A
s
, and the same forces. Also,
the total mass of each disk is the same.
(a) The torque about the disk center is
R
s
F
for the disks 1 and 2, and it is
R
c
F
for the disk 3. The
inner circle is smaller than the outer, so
.
3
,
2
,
1
,
com
com
com
τ
τ
τ
>
=
(b) The rotational inertia of disks 1 and 3 are the same by inspection. The rotational inertia of
disk 2 is smaller because it has more mass (the same area but denser material) closer to the
center of mass than either disk 1 or 2. Therefore,
I
1
= I
3
> I
2
.
(c) The angular acceleration is
τ
α
=
I,
so the larger the rotational inertia the smaller the angular
acceleration for the same torque. So, disk 2 has larger angular acceleration than disk 1. Also, the
larger the torque the larger the angular acceleration if the rotational inertia does not change, so
disk 1 has larger angular acceleration than disk 3. Therefore,
.
3
1
2
α
α
α
>
>
2. P.10.27. (a) Earth makes one rotation per day and 1
d
is (24 h) (3600 s/h) = 8.64
×
10
4
s, so
the angular speed of Earth is
5
4
2
rad
7.3
10
rad/s.
8.64
10 s
ω
−
π
=
=
×
×
(b) We use
v
=
ω
r
, where
r
is the radius of its orbit. A point on Earth at a latitude of 40° moves
along a circular path of radius
r = R
cos 40°, where
R
is the radius of Earth (6.4
×
10
6
m).
Therefore, its speed is
5
6
(
cos 40 )
(7.3 10
rad/s)(6.4
10 m)cos40
3.5
10 m/s.
v
R
ω
−
=
° =
×
×
° =
×
2
2
(c) At the equator (and all other points on Earth) the value of
ω
is the same (7.3
×
10
–5
rad/s).
(d) The latitude is 0° and the speed is
5
6
(7.3 10
rad/s)(6.4
10 m)
4.6
10 m/s.
v
R
ω
−
=
=
×
×
=
×

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