Three blocks are on an incline of Theta(a), block 1 has mass of m1, block 2 has mass of m2, and block 3 has mass of m3. Block 2 is being held(theoretically) above the ground by friction between block 1 and block 3. What is the minimum magnitude of force positioned directly parallel to the x axis on block 1 required to hold block 2 in the air between the two blocks? The ramp is frictionless and the coefficient of static frictions are us1 and us3 for block 1 and block 3 respectively.
Fcos(a) - F(n1) - m1gsin(a) = m1a
F(n1) - F(n2) - m2gsin(a) = m2a
F(n2) - m3gsin(a) = m3a
and for the friction
m2gcos(a) - us1F(n1) - us3[F(n2) + m2gsin(a)] = 0
The Attempt at a Solution
I am not sure if my friction equation is correct. I know the friction from the two blocks has to equal m2gcos(a) but I'm not sure if [F(n2) + m2gsin(a)] is correct and if it is, did I place the static friction coefficients backwards? I'm trying to visualize what is happening to that middle block and I feel like the gravity force should be on the first block therefore should use "us1" but that goes against my whole force diagram. So I think I dun goofed. I got an answer but that one point is making me feel that the answer is incorrect. Maybe I'm thinking about friction wrong? The weight force is in the direction of of the normal F(n2) force but since friction only accounts for normal forces then it actually adds to the other normal force?
F=(m2gcos(a)-us2m2gsin(a))/((us(m2cos(a)+m3cos(a))/(m1+m2+m3)) + (us2(m3cos(a))/(m1+m2+m3))