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Homework Statement
In the arrangement shown in fig 1.24, the masses of the bar, m, and of the wedge, M, and also the wedge angle, ##\alpha##, are known. The masses of the thread and pulley are negligible, and friction is absent. Find the acceleration of the wedge M.
The Attempt at a Solution
Let the positive x direction be towards the wall, and the positive y direction be the direction of gravity.
Let ##a_x## and ##a_y## be the respective components of mass m, and let ##a## be the magnitude of the acceleration of M. (Note: we expect ax to be negative.)
We can write three net-force-equations for the three accelerations:
##ma_y=mg-F_N\cos\alpha##
##ma_x=T\cos\alpha-F_N\sin\alpha##
##Ma=T(1-\cos\alpha)+F_N\sin\alpha##
Of course this introduces two more unknowns (the normal force FN between m and M, and the tension T in the thread).
So we need two more equations for the problem to be solved.
Conceptually, the two constraints that lead to these two equations are that m stays in contact with M, and that the length of the string is constant.
We can describe the fact that m stays in contact with M with the following equation:
##\frac{a_y}{a-a_x}=\tan\alpha##
Now about the constraint that the string length is constant... Well we can see the distance M moves towards the wall must be the distance m moves down the wedge. If we let ##\hat u## be the unit vector in the direction down the wedge's slope, and ##\hat x## and ##\hat y## be the unit vectors for the x and y directions respectively, then we can write this constraint as follows:
##a=(a_x\hat x+a_y\hat y)\cdot \hat u = a_x \hat x \cdot \hat u + a_y \hat y \cdot \hat u= a_y\sin\alpha-a_x\cos\alpha##So now we have five equations and five unknowns:
##ma_y=mg-F_N\cos\alpha##
##ma_x=T\cos\alpha-F_N\sin\alpha##
##Ma=T(1-\cos\alpha)+F_N\sin\alpha##
##\frac{a_y}{a-a_x}=\tan\alpha##
##a= a_y\sin\alpha-a_x\cos\alpha##
But upon solving I don't get the correct answer. I thought I was careful with the algebra.
Can anyone find any mistakes in these equations? (Particularly in the last two.) I can try to explain how I got any of them if requested.