Important Vector
SECTION 3.2 MOMENTS OF FORCE ABOUT A POINT
Example 3.2
Determine the moment of the 100-lb force F , shown in Fig. 3.11, about
points A and respectively.
As a first step, let us express force F vectorially. Note that the force
is collinear with the vector from to E , where
=
X i
4 j - 4kTo get a unit vectorp in the direction of we proceed as follows:
Figure 3.11. Find moments at A and
Xi
+
4 j 4k=
+
+
=
+
-We can then express force F in the following manner:
To get the moment about point A , choose a position vector
from point A to point which is on the line of action of force F. Thus, we have, for for we than =
+
4 j -,
40.8 -.
+
+
- - Therefore.-
245j+
ft-lhAs the moment about reference point E , we employ the position
vector from to position
D,
again on the line of action forceThus, we have = 4j - ft Accordingly, = F = (4j -
+
- =+
+
+
163.23-
-
326k3.1. What is position the 0, to the What its magnitude and direction cosines?
3.2.
to position 4) ft'?
3.3. A surveyor determines that the top a transmission
tower is at position =
+
her position. Similarly. the top of a tower is hy
=
+
+
m. What is the.he two tower tops?
3.4. Reference is rotated counterclockwise about its to reference is the position vector refer-
of a point having a position
as
What is the displacement vector from position
= hi'
+
+
i , and k primes) lor unit with refer-
1.5.
support A of the simply supported
Find the of ahout the support at A
1.6.
hen point
Find the moment of first point A and
Do u s e r vector approach.
N
Figure P.3.h.
3.7. A particle along circular path in plane. What the vector r this as of
Figure P.3.7.
3.8. A particle moves along a i n plane. If
particle has point = 4 j
+
give the pointFigure P.3.8.
An spotter on Hill high) the
cncmy 3.000 m NE h i m at a n position. A 105-nim with m spotter. and 155-mm
unit with a of m is SSE
Fig. Both gun units are located at an elevation
Can hoth hit an air in?
3.10. Find the of the forces about points A and (a) Use scalar approach.
Use vector approach.
Figure P.3.10.
3.11. The crew of a submarine patrol plane, with three-dimen-
sional radar, sights a surfaced submarine 10,000 yards north and yards east while flying at an elevation 3,000 ft above sea level. Where should the pilot a second patrol plane flying at an elevation of 4.000 ft at a position 40,000 yards east of the first plane to look for confirmation of the sighting?
3.12. A power company lineman can comfortably trim branches
I m from his waist at an angle of 45” above the horizontal. His waist coincides with the pivot of the work capsule. How high a
branch can he trim if the maximum elevation angle of the arm is and the maximum extended length is m?
1.5 m Figure P.3.12.
3.13. The total equivalent forces from water and gravity are
shown on the dam. (We will soon be able to compute such equiv- alents.) Compute the of these forces about the toe of the dam in the right-hand comer.
8 ’
Figure P.3.13.
3.14. In an underwater “village” research. an American flag is in place as shown. It is of plastic material and can rotate so as to be oriented parallel to the flow of water. A uniform friction force distribution from the flow is present on both faces of the flag hav- ing the N per square meter. Also the flagpole has a uniform force from the flow of 20 N per meter of length of the flagpole. Finally there is an upward buoyant force on the flag of
N and on the flagpole of N. What is the moment vector of these forces at the base of the flagpole?
Figure P.3.14.
3.15. transmission lines placed unsymmetrically on a power-line pole. For each pole, the weight of a single line when cov-
ered with ice is N. What is the moment at the base of a pole?
Figure P.3.15.
3.22. the moment of a IO-lh
3.17. A truck-mounted crane has a
the horizontal. What the moment due t o a
Do hy hy
Figure P.3.22. 3.23. Three guy w i r e s used in
that m A and R arc
a o f 30
Figure P.3.17. What i s wire the 0
with
3.18. A small i s shown in the
gram wherein and the centerline of A H arc A force F wind, weight. and shown
centerline the I t
=
+
+
what are the moment from F A .
Figure P.3.18.
68
3.24. Cables CD and help support member ED and the
I ,000-lb at D. At E there is a ball-and-socket joint which also
supports the member. Denoting the forces from the cables as
and respectively, compute moments of the three forces about
point E . Plane EGD is perpendicular to the wall. Get results in
terms of and
Figure P.3.24.
Moment of a
About an Axis
Case A. For Simple Cases.
By means of a simple situation, we shallset forth a definition of the moment of a force an axis. Suppose that a
disc mounted on a shaft that is free to rotate in a set of bearings, as shown in Fig. 1.12. A force F, inclined to the plane A of the disc, acts on the disc. We decompose the force into two coplanar rectangular components, one normal to plane A of the disc and one tangent to plane A of the disc, that is, into forces
and respectively, so as to form a plane shown tinted, normal to plane A.
We
Figure 3.12. turns disc
experience that does not cause disc
know from physics and intuition that the rotational motion of the disc is