Sunday, June 28, 2015

newtonian mechanics - How can an accelerating inclined plane prevent a block on it from sliding?


enter image description here


If I increase the force F, only the normal force N1 acting on m1 would increase which has no component along the plane, ie. along m1gsinθ, so how would applying this force prevent m1 from sliding?



When I view it from the accelerated frame of the wedge, enter image description here


It begins making sense. How is this possible? I'm really confused.



Answer



enter image description here


Diagram 1 shows the arrangement with the inclined plane stationary.
There are two forces acting on the block, its weight mg and the normal reaction on the block due to the inclined plane N1.
The resultant of theses two forces is F1(=mgsinθ) and this force accelerates the block down the slope.


Diagram 2 shows the situation when a force F is applied to the inclined plane and there is no relative movement between the block and the inclined plane.
That is because the resultant of the weight of the block mg and the now increased normal reaction N2 is a horizontal force F2.


If that force F2 on the block produces an acceleration of the block a which is the same as the acceleration of the inclined plane then the block will not move relative to the inclined plane.



When this condition is satisfied F2=ma and F=(m+M)a


Note that the magnitude of F controls the magnitude of N2 which in turn controls the direction of F2.


If F is larger than in the no relative movement condition then the magnitude of N2 is larger and the block accelerates up the inclined plane whilst if F is smaller then the block accelerates down the slope.


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