Two stacked blocks attached to a spring

[mintsnapple]

Homework Statement

Homework Equations

F_spring = kx, where x is the distance compressed
f_friction = uN

The Attempt at a Solution

Forces acting on m: N_1 up, mg down, f_friction to right
Forces acting on M: N_2 up, Mg+N_1 down, f_spring to the right

Block m will still be in equilibrium as long as the force exerted by the spring equals the force of friction:

kx = uN_1
The block isn't accelerating in y-direction so N_1 = mg
So kx = u*mg
x = u*mg/k

Does that look fine?

 

[Pythagorean]

Looks good to me, force balance between the friction and the spring gives the upper bound.

A small nitpick; I might have worded this differently:

"Block m will still be in equilibrium as long as the force exerted by the spring equals the force of friction:"

I would instead say "is equal to or less than".

 

[haruspex]

No, this is wrong. You've not taken into account that the spring also has to accelerate M, so the accelerating force on m will be less than kx.
Let the acceleration be a. Write out the horizontal ∑F=ma equations for both blocks. Don't forget that the horizontal friction force on m applies equally and oppositely to M.

A small nitpick; I might have worded this differently:

"Block m will still be in equilibrium as long as the force exerted by the spring equals the force of friction:"

I would instead say "is equal to or less than".

First, we're not asking for m to be in equilibrium - merely that it does not slip.
Secondly, the condition for not slipping is that the horizontal force required to accelerate it at the same rate as M does not exceed the maximum frictional force. The actual frictional force may be less than maximum.

 

[Pythagorean]

Ugh, thanks haruspex. I'm more disconnected from this stuff than I thought.

 

[HalfLostAndSearching]

Yes, your analysis and solution look correct. It is important to consider all the forces acting on the system in order to determine the equilibrium conditions. In this case, the force of friction must be equal to the force exerted by the spring in order for the system to remain in equilibrium. Your solution shows a clear understanding of these concepts.