What is the maximum force, P, before block A slips on block B?

[Maybe_Memorie]

Homework Statement

The two blocks shown in the diagram are at rest on a horizontal surface when a force P is applied to block B. Blaocks A and B have masses 20kg and 35kg respectively. The coefficient of friction between the two blocks is 0.35 and between the horizontal surface and Block B is 0.3.
Determine the maximum force, P, before A slips on B.

Homework Equations

F = ma , Friction max = uR

The Attempt at a Solution

Okay this thing has been killing me for months.
Friction between B and the ground is (20 + 35)g(0.3) = 16.5g and between B and A is
(0.3)(35)g = 10.5g. This produces anti-friction which propells A forward. Friction between A and B is (20)(0.3)g = 6g.
So, P should equal 16.5g +10.5g + (10.5g - 6g) = 31.5g N.
I'm supposed to be getting 35.75g N.

Where am I going wrong?

 

[Doc Al]

This produces anti-friction which propells A forward.

Anti-friction? The only force propelling A forward is the friction from B.

Hint: When the static friction between A and B is at its maximum value, what is the acceleration?

 

[Maybe_Memorie]

Anti-friction? The only force propelling A forward is the friction from B.

Hint: When the static friction between A and B is at its maximum value, what is the acceleration?

Sorry, that's what I mean.

The acceleration of A or B?
The acceleration of A relative to B will be 0 if slipping doesn't occur.

 

[Doc Al]

The acceleration of A or B?

Consider A first. What's the acceleration of A when slipping is just about to occur?

The acceleration of A relative to B will be 0 if slipping doesn't occur.

Exactly!

 

[Maybe_Memorie]

Well if the force pushing A forward is (10.5)g, the acceleration should be (10.5)g/20, =
(0.525)g m/s^2.

 

[Doc Al]

Well if the force pushing A forward is (10.5)g,

What's the maximum static friction force acting on A?

 

[Maybe_Memorie]

What's the maximum static friction force acting on A?

The normal reaction by the Coefficient of Friction. (20g)(0.3) = 6g N.

 

[Doc Al]

The normal reaction by the Coefficient of Friction. (20g)(0.3) = 6g N.

Good! Except that here μ = 0.35.

 

[Maybe_Memorie]

Good! Except that here μ = 0.35.

Okay, I did it again, and got the right answer.
So let me see if I've understood.

I've included a force diagram.

The frictional forces acting on B are [(55g)(0.3) + (35)(0.35)] = 28.75g N

The force pushing A forward is (20g)(0.35) = 7g N.

To make B move, P must be 28.75g, and to ensure that there is no relative acceleration between A and B it must be (28.75g + 7g) = 35.75g N.

 

[Doc Al]

The frictional forces acting on B are [(55g)(0.3) + (35)(0.35)] = 28.75g N

The friction force of A on B is equal and opposite to the friction force of B on A (which you have calculated correctly).

The force pushing A forward is (20g)(0.35) = 7g N.

Good. So what must be the acceleration of A?

 

[Maybe_Memorie]

The friction force of A on B is equal and opposite to the friction force of B on A (which you have calculated correctly).

If the friction force on B is the coefficient of friction by the normal reaction, should it not be
(0.35)(20g)?

Good. So what must be the acceleration of A?

F = ma, ma = 7g, so a = 7g/20 = 0.35g m/s^2.

 

[Doc Al]

If the friction force on B is the coefficient of friction by the normal reaction, should it not be
(0.35)(20g)?

Exactly.

F = ma, ma = 7g, so a = 7g/20 = 0.35g m/s^2.

Good! So what must be the acceleration of both A and B at that point? What force P is required to produce such an acceleration?

 

[Maybe_Memorie]

Good! So what must be the acceleration of both A and B at that point? What force P is required to produce such an acceleration?

If the acceleration of A is 0.35g, the acceleration of B must also be 0.35g, so that there is no relative acceleration between them and slipping will not occur.

F = ma, so P = (35)(0.35g).
Evidently I've gone wrong somewhere. Haha.

 

[Doc Al]

If the acceleration of A is 0.35g, the acceleration of B must also be 0.35g, so that there is no relative acceleration between them and slipping will not occur.

Good!

F = ma, so P = (35)(0.35g).
Evidently I've gone wrong somewhere. Haha.

P is not the only force acting on B.

Hint: Try treating both blocks as a single object.

 

[Maybe_Memorie]

P is not the only force acting on B.

Hint: Try treating both blocks as a single object.

P minus the two frictional forces acting on B has to produce an acceleration of 0.35g.

So [P - 16.5g-7g = 35(0.35g)]
P = 35.75g N

And if both blocks were a single object, the 7g N forces would not affect the system, and the mass of the system would be 55kg.

So [P -16.5g = 55(0.35g)]
P = 35.75g N

 

[Doc Al]

There you go.

 

[Maybe_Memorie]

Thank you very much for your help.