A block of mass m is held motionless on a frictionless plane of mass M and angle of inclination θ. The plane rests on a frictionless horizontal surface. The block is released. What is the horizontal acceleration of the plane?
(Introduction to Classical Mechanics by David Morin, question 3.8)
The Attempt at a Solution
N1 be the normal force exerted by the block on the plane and vice versa.
N2 be the normal force exerted by the ground on the plane and vice versa.
aM be the acceleration of the plane.
av be the vertical acceleration of the block.
ah be the horizontal acceleration of the block.
Considering the whole system:
N2 = Mg + mg
Considering the subsystem of the forces acting on the plane:
Vertical equilibrium: N2 = Mg + N1cosθ
Horizontal motion: N1sinθ = MaM
Considering the subsystem of the forces acting on the block:
Vertical motion: N1cosθ - mg = mav
Horizontal motion: N1sinθ = mah
M, m and θ are known. So I have 5 equations with 5 unknowns. However, solving the first and second equations yield: N1cosθ = mg , which, using the 4th equation will result in av = 0 and that's wrong for sure.
Looking at the answers, I see that my 3rd, 4th and 5th equations are correct, but the 1st and 2nd equations are not in the answer. So why is it that I can't use them (or why is it wrong)?