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 .