Page 66 - Dynamics of Mechanical Systems

0593_C02_fm Page 47 Monday, May 6, 2002 1:46 PM

Review of Vector Algebra 47

P2.4.7: See Figure P2.4.7. Forces F ,…, F act along the edges and diagonals of a parallel-
6
1
epiped (or box) as shown. Let i, j, and k be mutually perpendicular unit vectors parallel
to the edges of the box. Let the box be 12 inches long, 3 inches high, and 4 inches deep.
Let the magnitudes of F ,…, F be:
1
6
F = 15lb F = 18lb F = 18lb
1 2 3
F = 20lb F = 17 lb F = 26lb
4 5 6
a. Find unit vectors n ,…, n parallel to the forces F ,…, F . Express these unit vectors
1
6
1
6
in terms of i, j, and k.
b. Express F ,…, F in terms of unit vectors n ,…, n .
1
1
6
6
c. Express F ,…, F in terms of i, j, and k.
6
1
d. Find the resultant R of F . Express R in terms of i, j, and k.
1
e. Find the magnitude of R.
k
C H
4 in F 2
B G
F
6
F 3 in O D
1
F 4 j
A
FIGURE P2.4.7 12 in E F
3
Forces acting along the edges and diagonals
of a parallelepiped. i F 5
P2.4.8: Let forces be given as:
F = 3 n − 2 n + n N
1 1 2 3
F = 4 n + 6 n − 2 n N
2 1 2 3
F =− n + 3 n + 4 n N
3 1 2 3
F = 7 n − 8 n + 3 n N
4 1 2 3
F =− 5 n + 4 n − 9 n N
5 1 2 3
where n , n , and n are mutually perpendicular unit vectors.
3
2
1
a. Find a force F such that the resultant of F and the forces F ,…, F is zero. Express
1
6
F in terms of n , n , and n .
3
2
1
b. Determine the magnitude of F.
P2.4.9: See Figure P2.4.9. Let P and Q be points of a Cartesian coordinate system with
coordinates as shown. Let n , n , and n be unit vectors parallel to the coordinate axes as
x
y
z
shown.