The Theorem of Moments

It can be shown that for any force system the moment of the resultant about any point or axis is equal to the sum of the moments of the individual forces about that point or axis. We shall demonstrate this theorem for the special case of two co-planar, non-parallel forces.

Consider the two forces and shown in Fig. 1, with point of concurrence and moment-axis . Take for the -axis a line through and . Writing components along the -axis, we have: Multiply each term by the distance . From the geometry of Fig. 1, it will be seen that: so that: Since is the moment of about , is the moment of about , and the moment of the resultant about , the theorem for this case is proved.

If a moment vector is defined with respect to a point , then its component in any direction is the moment about the line . In Fig. 2, using the theorem of moments, the components of along the three coordinate axes can be determined by resolving the force into its rectangular components , , and and writing the sums of the moments of these components about the coordinate axes. This gives:

The signs of the terms have been fixed by the definition of a moment vector as previously given. If one looks from the origin towards the positive end of the coordinate axes, a clockwise rotation means a positive sign.

1.5.1 PROBLEMS

1. Prove the theorem of moments for a system of two parallel forces. The diagram of Fig. 1 of Section: Systems of Parallel Forces and the results of the accompanying analysis may be used.

2. Calculate the moment of the 1000-lb force about the point , (a) by multiplying the force by the perpendicular distance from to the line of action of the force, and (b) by resolving the force into rectangular components at the point , using the theorem of moments. (c) Repeat, by resolving the force into components at points and .

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3. is parallel to the -axis and is parallel to the -axis and the lines of action of the two forces intersect the -axis at a distance from the plane. Show that if the two vectors and representing the moments of and about are added, the resultant vector correctly represents the moment of the resultant of the two forces about the point .

4. Find the moment of the three forces shown in the figure about the axis . Do this by first finding the components of the moments along the three coordinate axes, and then by adding the components of these components along the line .

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5. Forces and are applied along the diagonals as shown in the figure. If lb, what is the moment sum about each coordinate axis? What is the resultant moment about ?

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, ; ,

6. Given lb, . The point 3, 4, 12 is a point on the line of action of . Find and .

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7. A plane force system is found to have: lb, lb, . Determine the point at which the resultant force intersects the -axis.

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8. The and components of a force are , and lb. The moments are , , . Locate the point of intersection of the force with the plane.

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ft, ft

1.5.1 PROBLEMS

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