fisica1-youn-e-freedman-exercicios-resolvidos

9.94: a) From the parallel-axis theorem, the moment of inertia is
2 2
I P
= (2 5) MR + ML , and
2
I P
⎛ 2
1
.
2
5 ⎟ ⎞
⎜ ⎛ ⎞⎛
R ⎞
= +
⎜ ⎟⎜
⎟
ML
⎝ ⎝ ⎠⎝
L ⎠ ⎠
2
2
If R = (0.05)L, the difference is 2 5)(0.05) = 0.001. b) ( I ML ) = ( m 3M
), which
is 0.33% when m = (0.01) M.
rod
(
rod
rod
9.95: a) With respect to O, each element
2
i
2 2
i
y i
r in Eq. (9.17) is x + , and so
2
2 2
2
2
∑miri
= ∑mi
( xi
+ yi
) = ∑mi
xi
+ ∑ mi
yi
= I
x
+
I =
I .
O
i i i i
b) Two perpendicular axes, both perpendicular to the washer’s axis, will have the
same moment of inertia about those axes, and the perpendicular-axis theorem predicts
M 2 2
that they will sum to the moment of inertia about the washer axis, which is ( R + ),
and so I
x
M
= = ( R + ).
I
y
2 2
4 1
R2
y
1
R
2 2
1 2
c) From Table (9.2), I = m(
L
2
2
+ L ) = 1
mL .
12
1 2
Since I
0
= I
x
+ I
y, and I
x
= I
y,
both I
x
and I
y
must be
12
mL .
6
M
9.96: Each side has length a and mass , and the moment of inertia of each side about
4
1
an axis perpendicular to the side and through its center is M 2
2
Ma
12 4
a =
48
. The moment of
inertia of each side about the axis through the center of the square is, from the
2
2
2
Ma M a Ma
+ = . The total moment of inertia is the sum of
perpendicular axis theorem, ( )
48 4 2 12
2
2
Ma Ma
the contributions from the four sides, or × = .
4
12 3