What is the Acceleration of an Inclined Plane due to Recoiling Block?

[WannabeNewton]

Homework Statement

A block of mass m is on an inclined plane of mass M, inclined at angle θ, and slides on the plane without friction. Find the acceleration of the plane.

The Attempt at a Solution

I am using the usual cartesian coordinate system with no rotations and letting up and forward be the positive directions. Let A be the acceleration of the plane as it accelerates backwards from the recoil due to the moving block. Define a non - inertial reference frame that is co - moving with the plane. The equations of motion for the block in this frame are $m\ddot{y} = Ncos\theta - mg$,$m\ddot{x} = F_{apparent} = Nsin\theta - mA$ and we have, in this co - moving frame, the constraint $\ddot{y} = -tan\theta \ddot{x}$. The equation of motion for the plane in this frame is $0 = F'_{apparent} = -Nsin\theta - MA$. Combining the equations for the block we get that $N = mgcos\theta + mAsin\theta$ so $MA = -Nsin\theta = -mgcos\theta sin\theta - mAsin^{2}\theta$ therefore $A = -(\frac{mgcos\theta sin\theta }{M + msin^{2}\theta })$. The book has the same answer except it is positive. I'm not sure why mine is negative. They don't really list if they are taking the backwards direction to be positive or not so I don't know if that is all there is to the issue. Thanks.

 

[oli4]

Hi Isaac
There is nothing in the statement that would give you a preferred direction, I read it as finding the magnitude of the acceleration of the plane. Otherwise, since the acceleration is a vector in the end, you could as well wonder about which component is which.
So I don't think there is any issue at all, you solved the problem :)

 

[haruspex]

Assuming you have exactly reproduced the statement of the problem from the book, I would guess you are supposed to take the positive direction as being whichever way the wedge moves. You say you let "up and forward be the positive directions", but you don't say forward for which object. If you meant forward for the block then you would expect a negative result for the wedge.
(Good job getting the right answer - easy to go wrong with a question like this.)