Page 173 - Fisika Terapan for Engineers and Scientists

12.2 Rotation about a Fixed Axis 373

move without slipping, the speed v of the ground must match the tangential
speed of the bottom point of the wheel; that is, v R. This proportionality
of v and implies the same proportionality of the accelerations a and , and
therefore establishes the relationship a R.
The angular acceleration of the wheel is then

a 3.7 m/s 2 2
12 radians/s
R 0.30 m

The large centrifuge shown in the chapter photo has an arm of
EXAMPLE 5 Concepts
length 8.8 m. When rotating at 175 revolutions per minute, in
Context
what is the speed of the end of this arm, and what is the centripetal acceleration?
SOLUTION: 175 rpm amounts to 175/60 2.9 revolutions per second. The
corresponding angular velocity is
2pf 2p 2.9 rev/s 18 radians/s

According to Eq. (12.11), the speed at a radius R 8.8 m is

2
v R 18 radians/s 8.8 m 1.6 10 m/s
and according to Eq. (12.15), the centripetal acceleration is

2
2
3
a R (18 radians/s) 8.8 m 2.9 10 m/s 2
centripetal
This is almost 300 standard g’s!

✔ Checkup 12.2

QUESTION 1: Consider a point P on the rim of a rotating, accelerating flywheel and
a point Q near the center. Which point has the larger instantaneous speed? The larger
instantaneous angular velocity? The larger angular acceleration? The larger tangential
acceleration? The larger centripetal acceleration?
QUESTION 2: The Earth rotates steadily around its axis once per day. Do all points
on the surface of the Earth have the same radius R for their circular motion? Do
they all have the same angular velocity ? The same speed v around the axis? The
same centripetal acceleration a ? If not, which points have the largest R, ,v,
centripetal
and a ?
centripetal
QUESTION 3: A short segment of the track of a roller coaster can be approximated by
a circle of suitable radius. If a (frictionless) roller-coaster car is passing through the
highest point of the track, is there a centripetal acceleration? A tangential accelera-
tion? What if the the roller coaster is some distance beyond the highest point?
QUESTION 4: Consider the motion of the hammer shown in Fig. 12.1. Taking into
account only the rotational motion, which end of the hammer has the larger speed v
around the axis? The larger centripetal acceleration a ?
centripetal
(A) Head end; head end (B) Head end; handle end
(C) Handle end; head end (D) Handle end; handle end