5.2 FORCE COUPLES TO MAKE CENTERLESS MOMENTS
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Sometimes we are faced with moments that are in awkward positions, We can move these by replacing them with forces and moments in new positions. (Force couples also allow us to rotate the effects of moments)
Keep in mind that couples cause rotation only, without any translation.
Consider the basic force couple,
A force couple acts about an axis. In the previous example the axis was out of the page. The axis can be moved to any position on the rigid body, as long as the direction is maintained. This axis of the couple will be perpendicular to the plane of the forces.
A sample problem is given using a rope pulling a wedge ([Hibbeler, 1992], prob 4-81, pg. )
5.2.1 Moving Forces and Equivalent Force Moments
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If a force is moved along the line of action, there are no other modifications needed, (this is called transmissibility)
If a force is not moved along its axis (line of action), then we must create a resultant (make believe) force couple (and it's resulting moment).
A sample of a problems that is simplified by moving forces is seen below, ([Hibbeler, 1992], prob 4-102, pg. )
5.2.2 Practice Problems
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1. The pole OA is 5m long and is held firm at base O. It is pulled by two cables AC and AB that each have 1KN of tension. Find the magnitude of the moment at O caused by the two cables
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2. A 500 lb cylindrical tank, 8ft in diameter, is to be raised over a 2ft obstruction. A cable is wrapped around the tank and pulled horizontally as shown. Knowing the corner of the obstruction at A is rough so that the cylinder slips at C but not at A, find the required tension in the cable.
3. A force of 1200N acts on a bracket as shown. Determine the moment Ma of the force about A.
4. A 30lb force acts on the end of the 3ft lever as shown. Determine the moment of the force about O.
5. Considering moments, convert the cases given on the left to those requested on the right.
6. Given the two forces F1 and F2 acting on point A and the force of gravity acting on the centre of the beam Fg, write the equation for the sum of the moments about C.
7.The frame below has a ball joint at A, and the other end of the beam at C is smooth and slides freely along the y-axis. A mass of 10kg is added midway along the beam AC. Find the reaction forces at A and C.
8. Find the moment at A given the two forces at B and C.
9. A box is shown in the figure below. Find the moment that the tension cable causes about the origin using a cross product. Give the results using components and a magnitude.
10. Find the tension in the cable DE.
11. A force P acts on a corner of a frame (at A) and creates a moment about the origin (O). We know that the force P is in the y-z plane, the moment about the y-axis is -20Nm, and the moment about the z axis is -40 Nm. Find the magnitude of the moment about the x-axis.
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12. Given the force at C and couple at B, find the magnitude and direction of the reaction forces at the supports A and E below.
5.2.3 References
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Beer, F.P., Johnson, E.R., Statics & Mechanics of Materials, McGraw-Hill, 1992.
Hibbeler, R.C., Engineering Mechanics: Statics and Dynamics, 6th edition, MacMillan Publishing Co., New York, USA, 1992.
