Page 140 - Dynamics of Mechanical Systems

0593_C04*_fm Page 121 Monday, May 6, 2002 2:06 PM

Kinematics of a Rigid Body 121

where t is time and n , n , and n are mutually perpendicular unit vectors fixed relative
3
2
1
to the box as shown. The velocity and acceleration of corner A of the box in R are:
R A
V = 2 n + 5 n − 6 n ft sec
1 2 3
and

R A 2
a = 2 n − 8 n − 4 n ft sec
1 2 3
Find at times t = 0 and t = 1:

a. The angular acceleration of the box in R.
b. The velocity of corner B in R.
c. The acceleration of corner A in R.

Section 4.10 Points Moving on a Rigid Body
P4.10.1: A particle P moves along a diametral slot of a rotating disk D as in Figure P4.10.1.
Let the speed v of P be constant at 2 ft/sec and directed from A to B. Let the angular
speed ω and angular acceleration α of D be 3 rad/sec and 5 rad/sec , respectively, with
2
directions as indicated in Figure P4.10.1. Finally let the radius of D be 2 ft. Find the velocity
and acceleration of P, relative to a fixed reference frame R in which D rotates, when P is
at (a) A, (b) O, and (c) B. Express the results in terms of the radial and tangential unit
vectors n and n .
θ
r
n
n θ θ n r
n
r
D B A Q P B
P ω = 3 rad/sec 30° 30° ω = 3 rad/sec
O
D O
2 ft
2 ft
A α = 5 rad/sec 2 α = 5 rad/sec 2

FIGURE P4.10.1 FIGURE P4.10.2
A particle moving along a diametral slot of a A particle P moving along a chord slot of a rotating
rotating disk. disk.

P4.10.2: Repeat Problem P4.10.1, where the slot AB is now along a chord of the disk D as
in Figure P4.10.2. As before, let the speed of P be uniform at 2 ft/sec and directed from
A to B. Find the velocity and acceleration of P, relative to a fixed reference frame R in
which D rotates, when P is at (a) A, (b) B, and (c) Q, the midpoint of AB. Express the
results in terms of the radial and tangential unit vectors n and n .
θ
r
P4.10.3: See Problem P4.9.8. The box shown in Figure P4.10.3 has an angular velocity
relative to a reference frame R given by:

ωω = t n +(2 + t)n +(4 + t )n
R Box 2 2
1 2 3