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The block of mass $m$ slides on a wedge of mass $m$ which is free to move on the horizontal ground. Find the acceleration of wedge and block.

Sol.

Let $a$ acc. of wedge and $b$ acc. of block w.r.t. wedge. Applying Newton Laws of motion in $x$ direction,

$F_x=ma_{1x}+ma_{2x}=0$

My problem:

But why $F_x=0$, if it is then how can a system can have acceleration in $x$ direction, they not massless too?

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This situation calls for momentum conservation.

$a_{COM}=\frac{m_1a_1 + m_2a_2.....m_na_n}{m_1+m_2.....m_n}$.

Initially the acceleration of COM was $zero$ $(because\space a_1=a_2=0)$ and since there are no external forces involved, according to the law of conservation of momentum, the final acceleration of COM will also be $zero$.

Only external forces can change the acceleration of the COM and they are absent in this case.

For momentum to be conserved, the wedge and the block will have to move in opposite directions.

Therefore,

$\frac{m_1 a_1 + m_2 a_2}{m_1 +m_2}=0$,

$m_1a_1+m_2a_2=0$ $\tag1$

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Even though the only force acting on the two masses is gravity, in the downward direction, and the wedge is free-sliding (frictionless) in the x direction, there IS friction between the wedge and the block - and that frictional force is parallel to the slanted surface of the wedge (not straight down). This force has an x-component, which drives the wedge away from the block as it's sliding down in the opposite direction.

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Consider 3 systems representing the situation

A: whole system including both the wedge and the block.

B: only the wedge and

C: only the block.

If you consider the system A there is no net force in x direction(assuming no friction bw wedge and ground)

Now consider the other 2 systems B and C. In each case there is net force acting on both systems B and C which causes the acceleration. However the $ F_x$ on these systems will be equal and opposite in nature. There will be no displacement of the centre of mass in the x direction.

The $F_x$ of both these systems come from the NORMAL FORCE they exert on each other. As mentioned these forces are equal and opposite to each other making the net force on system zero. Drawing the free body dig. will help.

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