SqueeSpleen
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Homework Statement
We have an inclined plane with a mass ##M## and an angle ##\alpha## and a box of mass ##m## over it.
Everything is at the instant 0 (it's a problem of static, no dynamics).
a) What's the acceleration in the component x of the box?
b) What's the acceleration in the component y of the plane?
c) What's the acceleration in the component x of the box?
Homework Equations
Newton laws, momentum conservation I guess.
The Attempt at a Solution
I have tried but I'm failing.
The solutions are supposed to be
a)
$$
a_{1x}
\dfrac{ -m_{2} \cdot g \cdot \tan (\alpha) }{ m_{2} \sec^{2} (\alpha) + m_{1} \tan^{2} (\alpha) }
$$
b)
$$
a_{2x}
\dfrac{ m_{1} \cdot g \cdot \tan (\alpha) }{ m_{2} \sec^{2} (\alpha) + m_{1} \tan^{2} (\alpha) }
$$
c)
$$
a_{2x}
\dfrac{ -( m_{1} +m_{2} ) \cdot g \cdot \tan^{2} (\alpha) }{ m_{2} \sec^{2} (\alpha) + m_{1} \tan^{2} (\alpha) }
$$
I've tried several times, and the best I have got for a) was
$$
a_{1x}
\dfrac{ -m_{2} \cdot g \cdot \tan (\alpha) }{ m_{2} \sec^{2} (\alpha) + m_{1} \sec^{2} (\alpha) }
$$
I'm surely missing something.
The reasoning beggings as follows:
The magnitude of the force of the box over the inclined plane is ##g m_{1} \sin (\alpha )## as is the normal component of the gravity. That force as a direction ## (\sin( \alpha ), - \cos(\alpha))##.
Here is where I got confused. I know if the ramp has infinite mass, or is "glued" to the floor which to this porpuse is the same, the box would only have left a the tangent component of the force.
The thing is, I'm not sure how acceleration gets distributed between two masses.
I remember that in the case of colliding "balls" it was something like
$$
\dfrac{ m_{1} }{ m_{1}+m_{2}}
$$
and when assuming that is when I almost got the result, but I had ##\sec## in both places, no ##\tan##. So I guess I'm missing a ##\sin##. Any one knows where can I read theory to help me understand this a bit better and solve this problem? I guess if I can start playing with Newton Laws until I get the correct result, but I would like to have a better understanding.
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