Friction problem for two block system on an inclined plane

[cooldudeachyut]

Homework Statement

Two block M₁ and M₂ rest upon each other on an inclined plane. Coefficient of friction between surfaces are shown. If the angle θ is slowly increased, and M₁<M₂ then

Options :
1- Block A slips first.
2- Block B slips first.
3- Both slip simultaneously.
4- Both remain at rest.

Homework Equations

Taking frictional force between the two blocks as f₁ and between the block B and inclined plane as f₂, the equations for limiting values f_1max and f_2max :

f_1max = μ₂M₁gcosθ
f_2max = μ₂(M₂+M₁)gcosθ

The Attempt at a Solution

For Block A :
I only considered the case where f₁ reaches its limiting value, hence I get this inequality as the condition when block A may start slipping,

M₁gsinθ - μ₂M₁gcosθ ≥ 0

Which is equivalent to,
tanθ ≥ μ₂

For Block B :
Again, I only considered the case where f₂ reaches its limiting value but cannot figure out what's the magnitude/direction of f₁ on this block. So I assumed f_1max to act on this block in the direction up the slope as this basically provides least resistance and also complies with block A's case which may as well be the "limiting factor" for the case where block B slips, giving my inequality as,

M₂gsinθ - μ₂(M₂+M₁)gcosθ + μ₂M₁gcosθ ≥ 0

Which is again equivalent to,
tanθ ≥ μ₂

I'm confused how to proceed now as I think both blocks should start slipping simultaneously however the answer provided is option 2, i.e., block B will slip first.

 

[gneill]

Suppose they both are on the threshold of slipping at the same angle as you've found. If both blocks begin to move, will there be any relative motion between them? How might you check for this?

 

[Chestermiller]

Homework Statement

Two block M₁ and M₂ rest upon each other on an inclined plane. Coefficient of friction between surfaces are shown. If the angle θ is slowly increased, and M₁<M₂ then
View attachment 224984
Options :
1- Block A slips first.
2- Block B slips first.
3- Both slip simultaneously.
4- Both remain at rest.

Homework Equations

Taking frictional force between the two blocks as f₁ and between the block B and inclined plane as f₂, the equations for limiting values f_1max and f_2max :

f_1max = μ₂M₁gcosθ
f_2max = μ₂(M₂+M₁)gcosθ

The Attempt at a Solution

For Block A :
I only considered the case where f₁ reaches its limiting value, hence I get this inequality as the condition when block A may start slipping,

M₁gsinθ - μ₂M₁gcosθ ≥ 0

Which is equivalent to,
tanθ ≥ μ₂

For Block B :
Again, I only considered the case where f₂ reaches its limiting value but cannot figure out what's the magnitude/direction of f₁ on this block. So I assumed f_1max to act on this block in the direction up the slope as this basically provides least resistance and also complies with block A's case which may as well be the "limiting factor" for the case where block B slips, giving my inequality as,

M₂gsinθ - μ₂(M₂+M₁)gcosθ + μ₂M₁gcosθ ≥ 0

Which is again equivalent to,
tanθ ≥ μ₂

I'm confused how to proceed now as I think both blocks should start slipping simultaneously however the answer provided is option 2, i.e., block B will slip first.

I get the same answer you get.

 

[cooldudeachyut]

Suppose they both are on the threshold of slipping at the same angle as you've found. If both blocks begin to move, will there be any relative motion between them? How might you check for this?

At that angle there should be no relative motion between the blocks initially as they both receive same acceleration but I don't know how to calculate for later instants as f₁'s direction/magnitude bothers me.

So does that mean block A actually doesn't slip first because slipping is only considered relative to block B? Or does that mean both start slipping at the same time?

I get the same answer you get.

I see, option 2 is wrong after all.

 

[haruspex]

I see, option 2 is wrong after all.

It is suspicious that both coefficients are labelled μ₂. Looks like a cut-and-paste error in the diagram, and one of them should have had a different value.

 

[cooldudeachyut]

It is suspicious that both coefficients are labelled μ₂. Looks like a cut-and-paste error in the diagram, and one of them should have had a different value.

That makes sense, there must've been a different μ₁ coefficient between both blocks in the original problem.

 

[gneill]

So does that mean block A actually doesn't slip first because slipping is only considered relative to block B?

That would be my interpretation of "slipping", yes.

 

[haruspex]

That would be my interpretation of "slipping", yes.

Ok, but I don't think that corresponds to reality.
In the real world, kinetic friction is always less than static, and there is no simultaneity. One will slip first, and as soon as that happens there is less tendency to slip at the other interface. So only one block will slip, but if the coefficients are the same we cannot say which.