Page 364 - Dynamics of Mechanical Systems

0593_C10_fm Page 345 Monday, May 6, 2002 2:57 PM

Introduction to Energy Methods 345

where x, y, and z are coordinates, measured in meters, relative to an X, Y, Z Cartesian
system. Acting on P is a force F given by:

F = 4 n − 6 n + 8 n
x y z
where F is measured in Newtons, and n , n , and n are unit vectors parallel to the X-, Y-,
z
y
x
and Z-axes. Compute the work done by F in moving P from (0, 0, 0) to (2, 4, 8).
P10.2.2: The magnitude and direction of a force F acting on a particle P depend upon the
coordinate position (x, y, z) of F (and P) in an X, Y, Z coordinate space as:

22
2 2
2
F = 2xy z n + 2x yz 2 n + 2x y z n N
x y z
where n , n , and n are unit vectors parallel to X, Y, and Z. Suppose P moves from the
z
x
y
origin O to a point C (1, 2, 3) along two different paths as in Figure P10.2.2: (1) along the
line segment OC, and (2) along the rectangular segments OA, AB, BC. Calculate the work
done by F on P in each case. (Assume that the coordinates are measured in meters.)
Z
C(1,2,3)
n
z
P
n
y
O
Y
A B
P
FIGURE P10.2.2
A particle P moving from O to C along n x
X
two different paths.
P10.2.3: See Problem P10.2.2. Suppose a force F acting on a particle P depends upon the
position of F (and P) in an X, Y, Z space as:

∂φ ∂φ ∂φ
φ
F =∇ = n + n + n z
D
x
y
∂x ∂y ∂z
Show that the work done by F on P as P moves from P (x , y , z ) to P (x , y , z ) is simply
2
1
2
1
2
1
1
2
φ (x , y , z ) – φ (x , y , z ). Comment: When a force F can be represented in the form F =
1
1
1
2
2
2
∇φ, F is said to be conservative.
P10.2.4: See Problems P10.2.2 and P10.2.3. Show that the force F of Problem P10.2.2 is
conservative. Determine the function φ. Using the result of Problem P10.2.3, check the
result of P10.2.2.
P10.2.5: A horizontal force F pushes a 50-lb cart C up a hill H which is modeled as a
sinusoidal curve with amplitude of 7 ft and half-period of 27 ft as shown in Figure P10.2.5.
Assuming that there is no frictional resistance between C and H and that F remains directed
horizontally, find the work done by F.