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A wedge of mass m_{1} and angle alpha rests on a horizontal surface. A block of mass m_{2} is placed on the wedge, which rests on the wedge. Assuming friction to be negligible, find the acceleration of the wedge. Also find the acceleration of the block relative to the wedge.

Dear Student,

Please find below the solution to the asked query:

Before doing this problem, we look at two remarks first.

If M is not moving then we have the trajectory sketched in the right.

If M is moving to the left, then the actual trajectory is sketched as below, where *a _{M}* is the acceleration of the block relative to the wedge of mass M.

Now, we look at the force diagram of the wedge and the block.

$ForM:{N}_{2}\mathrm{sin}\theta =M{a}_{1}..............\left(1\right)\phantom{\rule{0ex}{0ex}}Form:mg-{N}_{2}\mathrm{cos}\theta =m{a}_{M}\mathrm{sin}\theta ....................\left(2\right)\phantom{\rule{0ex}{0ex}}{N}_{2}\mathrm{sin}\theta =m\left({a}_{M}\mathrm{cos}\theta -{a}_{1}\right)...................\left(3\right)$

This is a set of simultaneous equations with 3 unknowns. We obtain,

${a}_{1}=\frac{mg\mathrm{sin}\theta \mathrm{cos}\theta}{M+m{\mathrm{sin}}^{2}\theta}and{a}_{M}=\frac{\left(M+m\right)g\mathrm{sin}\theta}{M+m{\mathrm{sin}}^{2}\theta}$

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