PHYS310_homework_11_solutions - Homework set 11 due...

Homework set 11, due Wednesday, November 16, 11:30 am
Following problems from Analytical Mechanics by Fowles & Cassiday
1. 8.12
cm
mx mg T = - &&
cm
I aT ϖ =
&
cm
x a ϖ =
&& &
2
2
5
cm
I ma =
2
1 2 2
5 5
cm cm
cm cm
I x
mx mg mg ma mg mx
a a a
ϖ
 
= - = - = -
 
 
& &&
&& &&
5
7
cm
x g = &&
2. 8.20 (20 points)
mx mg μ =
&&
x g μ = &&
x gt μ =
&
, and
2
1
2
x gt μ =
2
2
5
I ma mga ϖ ϖ μ = = -
& &
5
2
g
a
μ
ϖ = - &
5
2
g
t
a
μ
ϖ ϖ = -
o
Slipping ceases to occur when
v a ϖ =
…
5
2
gt a gt μ ϖ μ = -
o
2
7
a
t
g
ϖ
=
o
2
1 2
2 7
a
x g
g
ϖ
μ
μ
 
=
 
 
o
2 2 2
49
a
x
g
ϖ
μ
= o
3. 8.22 (20 points)
From section 8.7 (see Figure 8.7.1), the instantaneous center of rotation is the
point
O
. If
x
is the distance from the center of mass to
O
and
2
l
is the distance
from the center of mass to the center of percussion O
′
, then from eqn. 8.7.10 …
a
x
T
r
mg
r

2 2 1
2 12 12
cm
I l Ml l
x
M M
   
= = =
   
   
6
l
x =
4. 9.1 (20 points)
9.1 (a)
( 29
2 2
xx
I y z dm = +
∫
dm dxdy ρ = and
2
2 m a ρ =
( 29
2
2
0 0
0
y a x a
xx
y x
I y dxdy ρ
= =
= =
= +
∫ ∫
2
0
2
a
a y dy ρ =
∫
2
4
2
3