Examples in Chapter 9

9.25

A flywheel with a radius of 0.3 m starts from rest and accelerates with a constant angular acceleration of 0.6 rad/s2 . Compute the magnitude of

the angular accleration the radial acceleration The resultant acceleration of a point on its rim

a) At the startb) After it has turned through 600

c) After it has turned through 1200

Starting conditions

= 0.3 rad/s0 =0

r= 0.3 m

Equations to think about

tan

2

2 2 2tan

radial

radial

v r

a r

a r

a a a

0.3

0.6

r

At the start

tan

2

2 2 2tan

radial

radial

v r

a r

a r

a a a

0.3

0.6

r

2tan

2

2 2 2tan

2

0.3*0 0

0.3*0.6 0.18 m/s

0

0.18 m/s

radial

radial

v

a

a r

a a a

a

At 600

2 20 0

tan

2

2 2 2tan

2

radial

radial

v r

a r

a r

a a a

0

00 0

0.3

0.6

0

60 *180 3

r

2 2

tan

2

2 2 2

0 2*0.6* 1.2563

.18

1.256 *.3 .3769

.18 .3769

0.417 m/s

radial

a

a

a

a

At 1200

2 20 0

tan

2

2 2 2tan

2

radial

radial

v r

a r

a r

a a a

0

00 0

0.3

0.6

0

2120 *

180 3

r

2 2

tan

2

2 2 2

0 2*0.6* 1.2563

.18

1.256 *.3 .3769

.18 .3769

0.417 m/s

radial

a

a

a

a

9.46

A light flexible rope is wrapped several times around a hollow cylinder with a weight of 40 N and a radius of 0.25 m that rotates without friction about a fixed horizontal axis. The cylinder is attached is attached to the axle by spokes of a negligible moment of inertia. The cylinder is initially at rest. The free end of the rope is pulled with a constant force P for a distance of 5 m at which the end of the rope is moving at 6 m/s. If the rope does not slip on the cylinder, what is the value of P?

Some figurin’

W=RWhere R= ½ I

v=r or v/r=Initially, 0 =0 so R= ½ I (v/r)2

From I=m*r2=(40 N/9.8)*(.25)2=0.255W=F*d or P*(5 m)W=5PP= (1/2)*0.255*(6/.25)2/5=14.7 N

9.71

A vacuum cleaner belt is looped over a shaft of radius 0.45 cm and a wheel of radius 2.0 cm . The arrangement is shown below. The motor turns the shaft at 60 rev/s and the shaft is connected to a beater bar which sweeps the carpet. Assume that the belt doesn’t slip.

a) What is the speed of a point on the belt?

b) What is the angular velocity of the belt?

Some more figurin’

Part a) v=r* where r=0.45 cm= 60 rev/s *(2*radians/rev)=377 rad/sv=0.45*377=169 cm/s or 1.69 m/s

Part b) 2= 169 cm/s / 2 cm =84.8 rad/s

Hint on 9.72

The wheels are coupled so that there is the tangential velocity is constant so that

1/2 = r2/r1