Blocks A and B each have a mass m. Determine the largest force F which can be applied to B so that A will not move relative to B if the inclined angle is 45o. All the surfaces are smooth.

We require that aA = aB = a
For block A:
S Fy = 0 and Ncos45o - mg = 0
S Fx = max and Nsin45o = ma also a = gtan45o
for block B:
S Fx = max and F - Nsin45o = ma
F - mg tan45o = mg tan45o   F = 2mg tan45o = 2mg

So, the largest force that can be applied on block B is F = 2mg