Friction and tension in a string

[haruspex]

I respectfully disagree. I maintain that the upper surface would slip first. Suppose we applied a force of only 30 N, rather than 50 N. Because of the symmetry of the system (i.e., the top surface is in contact with m1, which is held in place by an inextensible wire; this makes the contact of m2 with m1 equivalent to the contact of m2 with the table top), the friction force with m1 will be the same as the friction force with the table top, namely 15 N. So, the total force will be supported by equal frictional forces at the two surfaces. Now, it will take only about 17.3 N for slip to start occurring at the upper surface. So, if the force were increased to 35 N and m2 were deformable, that would be enough for slip to start occurring at the interface with m1. (Of course, at the interface with the table, m2 would not be slipping because the critical force of about 51.9 N would not be exceeded there). The slippage at the top will occur with a frictional force determined by kinetic friction (12.3 N), until the deformational stiffness of m2 brings the deformation to a halt. This will result in a tension of about 12.3 N in the wire.

Chet

I think you misunderstood my point. I was looking at whether the mass can accelerate. The lowest total friction before that can happen is when the lower interface slips first (for whatever reason). 5*0.25+15*0.35 > 5*0.35+15*0.25. Yet even then it is greater than 50N. But now I realize there's an even simpler reason... the static friction between the lower mass and the ground is itself adequate. 15g*0.35 > 50. So I'm left puzzled as to what the author intended.

 

[Chestermiller]

We have no dispute as to whether the mass m2 can sustain significant movement; it can't. But what I'm saying is that, if we allow the mass m2 to be slightly deformable, the upper surface of m2 can move forward slightly (during which kinetic friction prevails), and we can thereby establish a value for the tension in the wire.

Chet

 

[haruspex]

We have no dispute as to whether the mass m2 can sustain significant movement; it can't. But what I'm saying is that, if we allow the mass m2 to be slightly deformable, the upper surface of m2 can move forward slightly (during which kinetic friction prevails), and we can thereby establish a value for the tension in the wire.

Chet

Certainly if the upper surface slips a little then the tension during the slip can be determined. Not entirely certain what happens when it stops again - maybe there's no reason for the tension to increase again at that point.
The conditions for the upper surface to slip are somewhat complex, involving not only the deformability of m2 but also of m1 and of the wire, and the relative heights of the blocks and points of attachment.

 

[Chestermiller]

The conditions for the upper surface to slip are somewhat complex, involving not only the deformability of m2 but also of m1 and of the wire, and the relative heights of the blocks and points of attachment.

I agree. I assumed for simplicity that the wire is inextensible and m1 is rigid. If either of these is relaxed, the problem becomes still more difficult.

Chet