- #1
- 5,844
- 551
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 [itex]m\ddot{y} = Ncos\theta - mg[/itex],[itex]m\ddot{x} = F_{apparent} = Nsin\theta - mA[/itex] and we have, in this co - moving frame, the constraint [itex]\ddot{y} = -tan\theta \ddot{x}[/itex]. The equation of motion for the plane in this frame is [itex]0 = F'_{apparent} = -Nsin\theta - MA[/itex]. Combining the equations for the block we get that [itex]N = mgcos\theta + mAsin\theta [/itex] so [itex]MA = -Nsin\theta = -mgcos\theta sin\theta - mAsin^{2}\theta [/itex] therefore [itex]A = -(\frac{mgcos\theta sin\theta }{M + msin^{2}\theta })[/itex]. 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.