Page 142 - Dynamics of Mechanical Systems

0593_C04*_fm Page 123 Monday, May 6, 2002 2:06 PM

Kinematics of a Rigid Body 123

P P α
Q Q
ω
30°
30°
D V n y
O O a O O V O ω n y
0.3 m n 0.3 m
x n
x
C C

FIGURE P4.11.1 FIGURE P4.11.2
A circular disk rolling on a straight line. A rolling, decelerating circular disk.
P4.11.3: An automobile A traveling at 30 mph goes around a curve as represented in Figure
P4.11.3. Consider the left rear wheel W . Let the radius of the curve, approximated as a
LR
circle, upon which W travels be 100 ft. If the diameter of W is 26 in., find the angular
LR
LR
velocity of W . Express the result in terms of unit vectors n , n , and n fixed relative to
z
x
LR
y
A, with n being forward, n left, and n vertical up.
x
y
z
n
x
n z
A
FIGURE 4.11.3 n
y
An automobile going around a turn
to the left.
P4.11.4: See Problem P4.11.3. Let the rear axle of A be 54 in. long. Repeat Problem 4.11.3
for the right rear wheel W .
RR
P4.11.5: See Problems P4.11.3 and P4.11.4. Suppose A is going around the curve at a
constant speed. Find the angular accelerations of W and W . As before, express the
RR
LR
results in terms of n , n , and n .
z
x
y
P4.11.6: See Problem P4.11.5. Suppose that instead of going at a constant speed around
2
the curve A is slowing at the rate of 8 ft/sec . Determine the angular acceleration of W LR
and W .
RR
P4.11.7: A circular disk D with radius r rolls on a circle of radius R with a lean angle
θ toward the inside of the circle as shown in Figure P4.11.7. Let n , n , and n be mutually
t
r
z
perpendicular unit vectors, with n being tangent to the circle at the contact point C
t
between D and the circle; n is a radial unit vector directed toward the center of the circle;
r
and n is a vertical unit vector perpendicular to the circle. Let the center O of D have a
z
velocity and tangential acceleration relative to a reference frame in which the circle is fixed
given by:
V = V n and a = a n
O
O
t t
If θ is a constant, determine expressions for the velocity and acceleration of point P at the
top of D as in Figure P4.11.7.