What is meant by "the body is on the point of ...." in mechanics?

[etotheipi]

I see this sort of wording a lot, for instance, we might say that the block is on the point of slipping or the ball is on the point of leaving the surface of the hill. My guess is that it's to do with constraint forces; that is, at the exact point where the constraint forces acting on a body can no longer constrain the motion of the body (perhaps because static friction has reached $\mu N$ or the normal contact force has reduced to zero), we say the body is on the point of doing something.

For instance, if a ball is rolling down a hill, there exists a normal contact constraint force (satisfying $N \geqslant 0$) which adjusts its magnitude so that the ball remains on the surface - the constrained motion. But when this force reaches $N=0$, if the velocity of the ball increases by $dv$ the normal force according to the constrained model would become negative - which evidently can't occur.

So my conclusion was that generally we write the equations of motion for the constrained motion (i.e. at rest, moving in a circle of fixed radius, moving with a platform etc.), and then substitute in the limiting case of a constraint force to solve for the conditions when the motion becomes unconstrained. Is this what is meant when we say something is on the point of e.g. slipping/toppling etc.?

I tried searching for references but the only mentions of constraint forces I could find were in the context of Lagrangian dynamics and other higher level mechanics. I don't know if the usage in that context is similar to what I've said above.

 

[PeroK]

That seems about right.

 

[etotheipi]

That seems about right.

Fair enough, thanks for your stamp of approval!

 

[berkeman]

we might say that the block is on the point of slipping

So my conclusion was that generally we write the equations of motion for the constrained motion (i.e. at rest, moving in a circle of fixed radius, moving with a platform etc.), and then substitute in the limiting case of a constraint force to solve for the conditions when the motion becomes unconstrained. Is this what is meant when we say something is on the point of e.g. slipping/toppling etc.?

I also think that you have a good grasp of what the term entails. There are other similar terms that I can think of -- "On the edge", "On the brink", "On the precipice" that all can be used to indicate the point where something changes in a situation that causes a change in motion or whatever.

Maybe one reason for the terminology of "point" or "edge" etc., is that if you plot the motion of the object versus time, there is an inflection in that plot at that "point". Certainly if you plot the position as a function of time of an object resting on a floor with friction as you linearly increase the horizontal force on it, that function will have an inflection point as the threshold force is reached to break it free and start moving it horizontally...