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## Homework Statement

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)

## Homework Equations

F=ma

## The Attempt at a Solution

Let:

N

_{1}be the normal force exerted by the block on the plane and vice versa.

N

_{2}be the normal force exerted by the ground on the plane and vice versa.

a

_{M}be the acceleration of the plane.

a

_{v}be the vertical acceleration of the block.

a

_{h}be the horizontal acceleration of the block.

Considering the whole system:

N

_{2}= Mg + mg

Considering the subsystem of the forces acting on the plane:

Vertical equilibrium: N

_{2}= Mg + N

_{1}cosθ

Horizontal motion: N

_{1}sinθ = Ma

_{M}

Considering the subsystem of the forces acting on the block:

Vertical motion: N

_{1}cosθ - mg = ma

_{v}

Horizontal motion: N

_{1}sinθ = ma

_{h}

M, m and θ are known. So I have 5 equations with 5 unknowns. However, solving the first and second equations yield: N

_{1}cosθ = mg , which, using the 4th equation will result in a

_{v}= 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)?