Block and wedge. Analysis of the center of mass

[clicwar]

Homework Statement

A block of mass m rests on a wedge of mass M and height H which, in turn, rests on a horizontal table. All the surfaces are frictionless.
If the system starts at rest with the block on the top of the wedge, find the displacement of the wedge during the time the block descends to the bottom.
Consider "alpha" as the inclination angle of the wedge related to the horizontal surface.

Homework Equations

Center of Mass equations and concepts

The Attempt at a Solution

Using pseudoforces i have first encountered the accelerations of the block and wedge, then the time taken by he block to slide down the wedge and then applying the kinematics equations to the wedge i encountered the displacement , which is D = H m cot("alpha")/(M+m)

BUT I'm sure that there is a easier method to solve this problem using the concept of center of mass(since there is no horizontal external force acting on the system the horizontal component of the center of mass position should be at rest, right?), but I'm stuck.
Could someone help me(or teach me) to find a easier way to get the answer without have to calculate all the accelerations as i have made in the long solution above.

Thanks in advance

 

[mjsd]

centre of mass(of the entire two objects system) in x direction before and after should remain at same place. so you need to find an x-adjustment to the wedge such that the new position combined with the block's new position will give you the same CM as the CM for the intial positions.
1. find initial CM by first work out location of CM for each mass: block and wedge
2. combine the two CMs to find the CM of system
3. work out final location of block
4. assume x amount of shift by wedge, combine with block's new position and demand it to equal to initial CM for system, and solve for x.

 

[m2003]

!

I appreciate your attempt at solving this problem using pseudoforces and kinematics equations. However, you are correct in thinking that there is a simpler approach using the concept of center of mass.

To solve this problem using the center of mass, we can first consider the system as a whole and find its center of mass. Since there are no external horizontal forces, the center of mass will remain at rest in the horizontal direction. This means that the displacement of the wedge will be equal to the displacement of the center of mass of the system.

Next, we can use the concept of conservation of energy to find the displacement of the center of mass. At the top of the wedge, the system has potential energy due to the height of the block and wedge above the ground. As the block slides down the wedge, this potential energy is converted into kinetic energy. At the bottom of the wedge, all of the potential energy will be converted into kinetic energy.

Using the equation for conservation of energy, we can set the initial potential energy equal to the final kinetic energy:

mgh = 1/2(mv^2 + MV^2)

Where h is the height of the wedge, v is the velocity of the block at the bottom of the wedge, and V is the velocity of the wedge at the bottom.

We can rearrange this equation to solve for the velocity of the block at the bottom:

v = √(2gh)

Now, we can use the definition of center of mass (xcm = (mx1 + Mx2)/(m+M)) to find the displacement of the center of mass:

xcm = (m(0) + M(Hsin("alpha")))/(m+M) = Hsin("alpha")/(1+m/M)

Therefore, the displacement of the wedge will be equal to the displacement of the center of mass, which is Hsin("alpha")/(1+m/M).

I hope this helps you understand the concept of center of mass and its application in solving this problem. Keep up the good work in your scientific pursuits!