What are the possible motions which can occur when you apply a force to an object sitting on a rough surface?
How do you determine which motion will occur in a particular situation?
Which types of motion are beyond the scope of Statics?
This section focuses on the various ways a rigid body in equilibrium might begin to move. The point at which an object starts to move is called the point of impending motion.
The interactive in Figure 9.2.1 shows a box sitting on a rough surface. Imagine that we start pushing on the side of the box with a gradually increasing force. Initially, friction between the block and the incline will increase to maintain equilibrium, and the box will sit still.
As we continue to increase the force there are two possibilities; the maximum static friction force will be reached and the box will begin to slide, or the pushing force and the friction force will create a sufficient couple to cause the box tip on its corner.
Instructions.
Free body diagram of a box on an inclined surface, pushed by a force \(A\text{.}\) The force, weight, box dimensions, friction coefficients, and slope are all adjustable. Under different conditions, the box will either tip, slip, or stay static. The force required for equilibrium is also shown.
Figure9.2.1.Slipping vs. Tipping
The easiest way to determine whether the box will slip, tip, or stay put is to solve for the maximum load force \(P\) twice, once assuming slipping and a second time expecting tipping, then compare the actual load to these maximums. This process is summarized in the following three steps:
Check slipping.
As in all dry friction problems, the maximum friction force is equal to the static coefficient of friction times the normal force
Assume that the maximum normal force \(N\) is acting at an unknown location and solve for the applied force which will maintain equilibrium. If the load exceeds this value than this the body will slip or maybe tip.
Figure9.2.2.FBD to check slipping.
Check tipping.
The object will tip when the resultant normal force \(N\) shifts off the end of the object, because it no longer acts on the object so it can’t contribute to equilibrium.
Create a free-body diagram assuming that the normal force \(N\) acts at the far corner of the box and solve for the applied force which will maintain equilibrium. Any greater force will make the body tip, unless it is already slipping.
At tipping, the friction force is static-but-not-impending as it has not reached impending motion for slipping.
Figure9.2.3.FBD to check tipping.
Compare the results.
If \(P\) exceeds the smaller of the limiting values, it will initiate the corresponding impending motion.
Thinking Deeper9.2.4.Failure in Engineering.
The goal of engineering design is to forecast and plan for all the ways that something can fail. The challenge is to know the questions to ask and the data to gather to model all possible failure modes. The controlling failure is the mode which occurs at the smallest load.
