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## Homework Statement

A particle of mass

*m*slides on the surface of a plane inclined at an angle

*θ*to the horizontal. The plane itself has a mass

*M*and is allowed to slide on a horizontal surface (such that

*θ*remains constant). There are no frictional forces.

(i) Write equations of motion for the particle and the inclined plane as viewed from an inertial reference frame.

(ii) By resolving the equations in (i) into a suitable coordinate system, find the acceleration of the particle and that for the inclined plane.

**2. The attempt at a solution**

Let

*N*be the reaction force of the plane on the particle and vice-versa (by NIII), and

_{1}*N*be the reaction of the horizontal plane on the inclined plane. If

_{2}**a**is the acceleration of the particle and

**A**is the acceleration of the plane:

[itex]m\mathbf{a}=\begin{pmatrix}ma_x\\ma_y\end{pmatrix}=\begin{pmatrix}-N_1\sin\theta\\N_1\cos\theta-mg\end{pmatrix}[/itex]

[itex]M\mathbf{A}=\begin{pmatrix}MA_x\\MA_y\end{pmatrix}=\begin{pmatrix}N_1 \sin\theta\\N_2-N_1\cos\theta-Mg\end{pmatrix}=\begin{pmatrix}N_1\sin\theta\\0\end{pmatrix}[/itex]

(see diagram - obviously forces are not drawn at correct positions on the plane, but this is irrelevant for the question anyway.).

It's part (ii) I'm having trouble with. Ignoring the y-acc of the plane, there are three equations and four unknowns (a

_{x}, a

_{y}, A

_{x}and N

_{1}). How is transforming to a different coordinate system going to help? Surely a fourth equation is needed. Any hints would be appreciated :)

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