Page 136 - Dynamics of Mechanical Systems

0593_C04*_fm Page 117 Monday, May 6, 2002 2:06 PM

Kinematics of a Rigid Body 117

P4.7.4: A sport utility vehicle traveling at 45 mph on a straight level roadway suddenly
goes out of control, spinning counterclockwise (looking from above) and becoming per-
pendicular to the direction of travel in 0.25 sec. If the wheel diameter is 26 in., determine
the angular velocity of the right rear wheel relative to the road surface. Express the result
in terms of unit vectors n , n , and n fixed in the vehicle, with n being forward and n z
y
z
x
x
being up. (Hint: The angular velocity of a wheel relative to the vehicle is equal to the
speed divided by the wheel radius.)
Section 4.8 Angular Acceleration

P4.8.1: Use Eq. (4.6.6) to validate Eq. (4.4.13).
P4.8.2: See Examples 4.7.1 and 4.8.1 and also Problem 4.7.1. Determine αα αα in terms of
R
D
unit vectors N , N , and N (i = 1, 2, 3) by using the result of Eq. (4.8.19) and the
Ri
Si
Di
configuration graph of Figure 4.7.6. Check the results by differentiating the expression for
R ωω ω ω as determined from Problem 4.7.1.
D
S
R
D
Y
Y
D S
P4.8.3: See Problem P4.8.2. Find αα αα , αα αα , and αα αα . Show that αα αα ≠ αα αα + αα αα + αα αα .
D
R
R
S
Y
D
S
Section 4.9 Relative Velocity and Relative Acceleration of Two Points on a Rigid Body
P4.9.1: A body B moves in a reference frame R with angular velocity in R given by:
R B
ωω= 6n − 4n + 7n rad sec
x y z
where n , n , and n are unit vectors parallel to an X-, Y-, Z-axis system fixed in B as in
x
y
z
Figure P4.9.1. Let O be the origin of X, Y, Z and let the velocity of O in R be:
R O
V =−10n x + 2n y − 8n ms
z
Z B
n P (1,4,3)
z
Y Q (2,8,4)
O n
y
FIGURE P4.9.1 X
A body B moving in a reference R n x
frame R.
Finally, let P and Q be particles of B with coordinates (1, 4, 3) and (2, 8, 4), respectively,
P
R
relative to X, Y, Z as shown. Find: (a) V ; (b) V ; (c) V P/Q . Check the results of (c) by
R
R
Q
constructing a vector r from Q to P and then evaluating dr/dt.
R
P4.9.2: See Problem P4.9.1. Let the angular acceleration of B in R be given by:
R αα= −2n x + 4n y − n rad sec 2
B
z