Page 168 - Fisika Terapan for Engineers and Scientists

368 CHAPTER 12 Rotation of a Rigid Body
y

rotation
axis
z P

R s

O x
rotating
fan O O y Angle = s /R
is measured
counterclockwise
from x axis.
P
FIGURE 12.4 The four blades x FIGURE 12.5 Motion of a reference particle P in
of this fan are a rigid body rotating Circular motion of a reference the rigid body rotating about a fixed axis.The axis is
particle P represents the angular
about a fixed axis, which coincides indicated by the circled dot O.The radius of the circle
orientation of the entire motion.
with the z axis. The reference par- traced out by the motion of the reference particle is R.
ticle P in this rigid body moves
along a circle around this axis.
letter phi) between the radial line OP and the x axis. Conventionally, the angle is
taken as positive when reckoned in a counterclockwise direction (as in Fig. 12.5). We
will usually measure this position angle in radians, rather than degrees. By definition,
the angle in radians is the length s of the circular arc divided by the radius R, or

s
angle in radians f (12.1)
R

In Fig. 12.5, the length s is the distance traveled by the reference particle from the x
axis to the point P. Note that if the length s is the circumference of a full circle, then
s 2pR, and f s R 2pR R 2p. Thus, there are 2 radians in a full circle;
that is, there are 2 radians in 360 :

2p radians 360
Accordingly, 1 radian equals 360 /2 , or

1 radian 57.3

The accuracy of the guidance system of the Hubble Space
EXAMPLE 1
Telescope is such that if the telescope were sitting in New York,
the guidance system could aim at a dime placed on top of the Washington
Washington, D.C. Monument, at a distance of 320 km. The width of a dime is 1.8 cm. What angle
does the dime subtend when seen from New York?
Length s of a small arc s ≈ 1.8 cm
segment is approximately
equal to dime‘s diameter. SOLUTION: Figure 12.6 shows the circular arc subtended by the dime.The radius
of the circle is 320 km. For a small angle, such as in this figure, the length s of the
arc from one side of the dime to the other is approximately the same as the length
of the straight line from one side to the other, which is the width of the dime.
R = 320 km
Hence the angle in radians is
New York angle subtended s 1.8 10 2 m 8
f 5.6 10 radian
5
R 3.2 10 m
Expressed in degrees, this becomes
FIGURE 12.6 A dime placed at a distance 8 360 6
of 320 km from the telescope. The length f 5.6 10 radian 3.2 10 degree
2p radians
s 1.8 cm is the diameter of the dime.