I was wondering if anyone could offer some help with the following question:
A wedge with mass M rests on a frictionless horizontal table top. A block with mass m is placed on the wedge. There is no friction between the block and the wedge. The system is released from rest.
Calculate the acceleration of the wedge.
Express your answer in terms of M, m,α and g.
The answer I should get is:
However I keep getting the following:
Does anyone see where I am going wrong? My working is as follows.
I will denote the normal force to the block by n.
Therefore the horizontal component of force on the wedge due to the block is given by F=-mgcosαsinα. The combined downward force due to the ramp and block is Mg+mgcosαsinα
Therefore (M+mcosαsinα)a=-mgcosαsinα (By Newtons second law)
I have no idea where I am going wrong so any help would be appreciated.
Consider the block and the wedge separately. The only forces acting on the block are its weight mg acting downwards, and the normal force n exerted by the wedge. The only forces acting on the wedge are its weight Mg acting downwards, a normal force equal and opposite to n (Newton's third law) exerted by the block, and an upwards normal force (not shown in the picture) exerted by the table.
The wedge slides towards the right with an acceleration a. The motion of the block is a combination of a component in the normal direction, together with a relative acceleration down the sloping face of the wedge.
Looking only at the wedge (not the block), the only horizontal force on it is the component of n. So the equation of motion in this direction is . (I seem to have reversed the sign of a somewhere along the line, so the acceleration is going to lose its negative sign.)
Now looking only at the block (not the wedge), the equation of motion in the normal direction says that the forces acting in that direction, namely , must produce the acceleration . (If the block does not have that component of acceleration in that direction then it will not stay in contact with the wedge.) So Newton's second law says that .
I don't know if that makes it any clearer. Apart from drawing the picture (and changing the sign of a), I seem to have more or less repeated my previous comment. I don't see what more I can give by way of explanation.
I was wondering if anyone could give me some hints as to how I am supposed to do part 3 of this question.
2.Calculate the horizontal component of the acceleration of the block.
Express your answer in terms of , , , and constant .
3.Calculate the vertical component of the acceleration of the block.Express your answer in terms of , , , and constant .
Note I have done part 2 and correctly got the answer ( I have added it to my post as I think it may be needed for part 3). The answer to part 2 is:
Anyway for part 3 I keep getting the wrong answer, could someone please explain how to do it?
By the way the answer you should get is:
But we know that and . So , and the equation of motion is
. Simplify that (using ) and you'll get the answer.