For mass y, I set my coordinate system to be the usual so:
W = -yg
T1 + W = 0, so T1 = yg
For mass x, I set my coordinate system so that +ve i is directed up the plane and +ve j is directed perpendicular to the plane (ie. the normal reaction)
N = Nj
F = Fi
W = -xg sin θ i - xg cos θ j
T2 = |T2| sin θj - |T2| cos θ i
I need to show that, to remain in equilibrium:
i) (mass) X >= Y tan θ
Since its a model pulled, the tension is the same on both sides. Hence, I figure this is something to do with the tension in the rope having to be less than the weight of X. Do I need to make the coordinate systems the same for both particles? If someone can point me in the right direction that would be helpful.
and ii) that μ >= (X sin θ + y cos θ) / (X cos θ - y sin θ)
My working so far is:
Resolving my equations in i gives |F| = |Xg| sin θ + |T2| cos θ
Resolving in j gives |N| = |Xg| cos θ + |T2| sin θ
And T1 = T2 = -yg (model pulley)
F <= u N
so u >= (Xg sin θ + yg cos θ) / (Xg cos θ + yg sin θ)
Obviously, we can divide each term by g and this is close, so I know Im going along the right lines, but the denominator should be (X cos θ - y sin θ) and Im not sure where my mistake is to make it a -ve.
Thanks for any help!