fisica1-youn-e-freedman-exercicios-resolvidos

2
10.67: For the disk, K = (3 4) Mv ( see Example10.6)
K = MgL sin , from which
1
β
. From the work-energy theorem,
L =
2
3v
=
4g
sin β
2
3(2.50 m s)
= 0.957 m.
2
4(9.80 m s ) sin 30.0°
This same result may be obtained by an extension of the result of Exercise 10.26; for the
disk, the acceleration is ( 2 3) g sin β,
leading to the same result.
b) Both the translational and rotational kinetic energy depend on the mass which
cancels the mass dependence of the gravitational potential energy. Also, the moment of
inertia is proportional to the square of the radius, which cancels the inverse dependence
of the angular speed on the radius.
10.68: The tension is related to the acceleration of the yo-yo by ( 2m ) g − T = (2m)
a,
and
a
to the angular acceleration by Tb = Iα = I
b
. Dividing the second equation by b and
adding to the first to eliminate T yields
2m
a = g
(2m
+ I
2
= g
2
b ) 2 + ( R
b)
2
,
2
α = g
2b
+ R
2
,
b
1 2 2
where I = 2 mR = mR
2
has been used for the moment of inertia of the yo-yo. The
tension is found by substitution into either of the two equations; e.g.,
T =
⎛ 2
2m)(
g − a)
= (2mg)
⎜1−
⎝ 2 + ( R
b)
2
⎞ ( R b)
⎟ = 2mg
⎠ 2 + ( R b)
2mg
=
.
(2( b R)
+ 1)
(
2
2
2