How Do You Solve the Block on Wedge Problem with a Moving Wedge?

[konichiwa2x]

Hi,
A block of mass 'm' slides on a wedge of mass M. The wedge also moves on a smooth frictionless surface. Find the acceleration of the wedge.

I have written down the equations of motion for the wedge and the block but don't know how to proceed further as there are more variables than equations:

a= ax + ay (block)
A = acceleration of wedge

For block:

$$Nsin\theta = m(ax)$$
$$Ncos\theta - mg = m(ay)$$

For wedge:
B is the normal reaction by the ground on Wedge
$$B = mg + Ncos\theta$$

What do I do after this? I think I have to work out a constraint relation here but I am not sure how to do it. Please help.

 

[Doc Al]

First find the horizontal accelerations involved. What's the relationship between A and ax? Your constraint is that the block maintains contact with the wedge. (Hint: View things from the frame of the wedge.)

 

[kyle8921]

I would suggest approaching this problem by using the principles of Newton's laws of motion. Firstly, we can consider the forces acting on the block and the wedge separately. For the block, we have the weight (mg) acting downwards and the normal force (N) acting perpendicular to the surface. The angle of the wedge (θ) also plays a role in determining the components of these forces.

For the wedge, we have the weight (Mg) acting downwards and the normal force (B) acting perpendicular to the surface. Again, the angle of the wedge (θ) will affect the components of these forces.

Next, we can apply Newton's second law to both the block and the wedge to determine their respective accelerations. For the block, we have:

m(ax) = Nsinθ

m(ay) = Ncosθ - mg

For the wedge, we have:

M(Ax) = Bsinθ

M(Ay) = Bcosθ - Mg

Since the block and the wedge are connected, there must be some constraint relation between their accelerations. This can be found by considering the fact that they are in contact with each other and therefore, their velocities must be equal. This can be expressed as:

ax = Ay

Solving these equations simultaneously will give us the acceleration of the wedge, A.

Finally, we can also use the principle of conservation of momentum to check our answer. Since there are no external forces acting on the system, the total momentum before and after the block and the wedge move must be the same. This can be expressed as:

(m+M)V = m(ax) + M(Ax)

Where V is the velocity of the block and the wedge after they start moving.

By solving these equations, we can determine the acceleration of the wedge and ensure that our answer is correct. I hope this helps!