Page 451 - Marks Calculation for Machine Design

P2: Sanjay
P1: Shibu/Rakesh
January 4, 2005
15:34
Brown.cls
Brown˙C10
U.S. Customary MACHINE MOTION SI/Metric 433
Step 2. Substitute the velocity (v A ) found in Step 2. Substitute the velocity (v A ) found in
step 1 and the given radius (r) of the rolling step 1 and the given radius (r) of the rolling
wheel in Eq. (10.51) to determine the angular wheel in Eq. (10.51) to determine the angular
velocity (ω) as velocity (ω) as
v A 88 ft/s v A 26.8 m/s
ω = = ω = =
rev 0.67 ft rev 0.2m
rad 1rev 60 s rad 1rev 60 s
= 132 × × = 134 × ×
s 2π rad 1 min s 2πrad 1 min
= 1,260 rpm = 1,280 rpm
From the principles of relative motion, the velocity of any other point on the wheel will
be the velocity (v A ), which has a magnitude of (rω), plus an additional velocity equal to
(rω) except directed perpendicular to the line connecting the point with the center of the
wheel and is in the direction of the angular velocity (ω). Fig. 10.22 shows the velocities of
three special points B, C, and D, and why the velocity of point P is in fact zero.
v
B B
w
v A rw
v
rw C
r
D
v A
C v A A v = rw
A
rw v D
rw v A
= 0
v P
FIGURE 10.22 Velocity of special points on a rolling wheel.
Therefore, the velocity at the top of the wheel, point B, has a magnitude

v B = v A + rω = v A + v A = 2 v A (10.52)
which is twice the velocity of the center of the wheel (v A ) and directed to the right as shown.
Also, the velocity of the instantaneous contact point P is zero as the velocity (v A ) to the
right is canceled by the velocity (rω) to the left.
The velocity (v C ) at the left side of the wheel, point C, has a magnitude given by the
pythagorean theorem as
√

2 2 2 2
v C = (v A ) + (rω) = (v A ) + (v A ) = 2 v A (10.53)
◦
and directed upward at 45 relative to the horizontal as shown.
Similarly, the velocity (v D ) at the right side of the wheel, point D, has a magnitude given
by the pythagorean theorem as
√

2 2 2 2
v D = (v A ) + (rω) = (v A ) + (v A ) = 2 v A (10.54)
◦
and directed downward at 45 relative to the horizontal as shown.