32.
(a) The angular speed in rad/s is
ω =
33
1
3
rev/min
2π rad/rev
60 s/min
= 3.49 rad/s .
Consequently, the radial (centripetal) acceleration is (using Eq. 11-23)
a = ω
2
r = (3.49 rad/s)
2
6.0
× 10
−2
m
= 0.73 m/s
2
.
(b) Using Ch. 6 methods, we have ma = f
s
≤ f
s, max
= µ
s
mg, which is used to obtain the (minimum
allowable) coefficient of friction:
µ
s, min
=
a
g
=
0.73
9.8
= 0.075 .
(c) The radial acceleration of the object is a
r
= ω
2
r, while the tangential acceleration is a
t
= αr. Thus
| a| =
a
2
r
+ a
2
t
=
(ω
2
r)
2
+ (αr)
2
= r
ω
4
+ α
2
.
If the object is not to slip at anytime, we require
f
s,max
= µ
s
mg = ma
max
= mr
ω
4
max
+ α
2
.
Thus, since α = ω/t (from Eq. 11-12), we find
µ
s,min
=
r
ω
4
max
+ α
2
g
=
r
ω
4
max
+ (ω
max
/t)
2
g
=
(0.060)
3.49
4
+ (3.49/0.25)
2
9.8
=
0.11 .