Page 450 - Marks Calculation for Machine Design

P2: Sanjay
P1: Shibu/Rakesh
January 4, 2005
15:34
Brown.cls
Brown˙C10
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APPLICATION TO MACHINES
Pulleys can also rotate and roll. The motion of simple to complex pulley arrangements
will be discussed, building on the discussions of both gears trains and rolling wheels.
10.4.1 Rolling Wheels
One of the most basic of motions in the study of machines is the velocity of a rolling wheel
on a flat surface, shown in Fig. 10.21.
w
(Geometric center)
r
A v A P(instantaneous contact point)

FIGURE 10.21 Velocity of a rolling wheel on a flat surface.

If the wheel rolls without slipping, then the velocity at point P is zero, and the velocity
of the geometric center of the wheel, point A, will be given by the expression
v A = rω (10.50)
where (r) is the radius of the wheel and (ω) is the angular velocity of the wheel. For the
clockwise angular rotation (ω) shown in Fig. 10.21, the velocity (v A ) of the center of the
wheel will be to the right as shown.
If the velocity (v A ) is known, which many times it is, then the angular velocity (ω) can
be found by rearranging Eq. (10.50) to give

v A
ω = (10.51)
r

U.S. Customary SI/Metric
Example 1. Determine the angular velocity Example 1. Determine the angular velocity
of a rolling wheel like that shown in Fig. 10.21, of a rolling wheel like that shown in Fig. 10.21,
where where
v A = 60 mph v A = 96.5 kph
r = 8in = 0.67 ft r = 20 cm = 0.2 m
solution solution
Step 1. Convert the given velocity of the center Step 1. Convert the given velocity of the center
of the rolling wheel to (ft/s) as of the rolling wheel to (m/s) as
mi 5,280 ft 1h km 1,000 m 1h
v A = 60 × × v A = 96.5 × ×
h mi 3,600 s h km 3,600 s
= 88 ft/s = 26.8 m/s