Volume of smaller pyramid is less than the volume of the whole.

[R_moor]

I have a test coming up next week and while doing some practice questions I found one I can't wrap my head around. The question is:

A pyramid (assume uniform density) is divided in two parts by a horizontal plane through its center of mass. How do the masses of the two parts compare ? There are three options provided:

a) The mass of the top part is higher
b) The mass of the bottom part is higher
c)Both parts have the same mass.

I think that for a symmetrical object like this one with uniform density the centre of mass must be equal in both sides. However, I'm not sure how to find a way to prove this, or (obviously) whether i am right or wrong. Any help Would be appreciated.

 

[haruspex]

symmetrical object

It may have some symmetries, but not a symmetry about the horizontal cut.
(You are not told how many sides the base has, or whether those sides form a regular polygon; but you do need to assume the base is horizontal.)
Where is the mass centre of a pyramid?
In geometrical terms, how does the whole pyramid compare with the portion above the cut?

 

[Alloymouse]

I have a test coming up next week and while doing some practice questions I found one I can't wrap my head around. The question is:

A pyramid (assume uniform density) is divided in two parts by a horizontal plane through its center of mass. How do the masses of the two parts compare ? There are three options provided:

a) The mass of the top part is higher
b) The mass of the bottom part is higher
c)Both parts have the same mass.

I think that for a symmetrical object like this one with uniform density the centre of mass must be equal in both sides. However, I'm not sure how to find a way to prove this, or (obviously) whether i am right or wrong. Any help Would be appreciated.

Consider this:

The centre of mass is typically below the halfway point of the height, since the base occupies a larger volume (and this object is of uniform density). I can't recall that generic number we use to find the height of the CM relative to the height of a uniform-density (and I am assuming square based) pyramid, but using that you can find the volumes of the top half and bottom half (or at least the proportions to the original mass M).

 

[R_moor]

Consider this:

The centre of mass is typically below the halfway point of the height, since the base occupies a larger volume (and this object is of uniform density). I can't recall that generic number we use to find the height of the CM relative to the height of a uniform-density (and I am assuming square based) pyramid, but using that you can find the volumes of the top half and bottom half (or at least the proportions to the original mass M).

Thank you this is what I did, I found the Centre of mass and noticed that there were two simple geometric figures left, A smaller pyramid at the top, and a trapezoid at the bottom.
Since: Mass = density * volume
and the density of both parts are the same, then the mass of each slice should depend on the volume,(at leasdt how they compare to each other.

 

[R_moor]

It may have some symmetries, but not a symmetry about the horizontal cut.
(You are not told how many sides the base has, or whether those sides form a regular polygon; but you do need to assume the base is horizontal.)
Where is the mass centre of a pyramid?
In geometrical terms, how does the whole pyramid compare with the portion above the cut?

For this I meant that you could have a vertical axis pass through the CM of mass, which will pass through the tip of the pyramid.

 

[haruspex]

I found the Centre of mass

What did you get?

A smaller pyramid at the top

How would the volume of that compare to the volume of the whole?