Consider a square ABCD of side 10cm and total mass 10kg. Let O be the center of the sqaure. We cut a square of corners O and C.

Find the central mass of the obtained shape after removing the small cutten sqaure.

How big is the cut-off piece? You would need to that it most cases. (Not this time, however)

I think they want the Center of Mass, not the "Central Mass" -- whatever that is.

You also probably meant "off corners A and C" instead of "of corners O and C". There IS no corner at O. Please be more precise with your problem copying.

In this case, because of the symmetry of removing opposite corners A and C, the center of mass will not move.

Lets do it the math way.

cmenw*massnew=cmold*oldmass-cmsquare*masssquare

now, doing it in each coordinate.

xcmnew*7.5=5*10-7.5*(10/4)=50-18.75=31.25
xcmnew=4.166

ycmnew*7.5=5*10-2.5*10/4
=50-6.25=43.75
ycmnew=5.833

check all that.

To find the center of mass of the shape, we need to determine the position of the shape's center of mass along both the x and y axes.

First, let's determine the position of the shape's center of mass along the x-axis.

The original square ABCD of side 10 cm and total mass 10 kg has a uniform mass distribution. Therefore, its center of mass coincides with the center of the square, O.

When we cut out the square of corners O and C, we are essentially removing a small square from the larger square. This small square will have a mass, let's call it M.

Since the small square is symmetrically removed from the larger square, the center of mass of the remaining shape will still lie at point O. Therefore, the position of the shape's center of mass along the x-axis remains unchanged.

Now, let's determine the position of the shape's center of mass along the y-axis.

The original square ABCD has a side length of 10 cm, so its center, O, lies halfway along the y-axis. Therefore, the position of the shape's center of mass along the y-axis is initially at y = 0 cm.

When we cut the square of corners O and C, the remaining shape will have a total mass of 10 kg - M.

Since the remaining shape is symmetric about the y-axis, the center of mass of the remaining shape will lie on the y-axis.

To find the position of the shape's center of mass along the y-axis, we can consider the mass balance equation:

M * (distance of the removed square's center of mass from the y-axis) = (10 kg - M) * (distance of the center of mass of the remaining shape from the y-axis)

Since the removed square is a square with its corners at O and C, the distance of its center of mass from the y-axis is 5 cm.

Since the remaining shape is symmetric about the y-axis, the center of mass of the remaining shape will also be at 5 cm from the y-axis.

So, the position of the shape's center of mass along the y-axis is y = 5 cm.

Therefore, the central mass of the obtained shape after removing the small square is located at point (x, y) = (0 cm, 5 cm).