A square aluminum plate (volume mass ρ = 2700 kg/m^3) has dimension L = 4.8 m and e = 0.9 m. A cylinder of radius R = 1.5 m and at a distance d = 1.2 m from the center of the plate and at 20 deg with the horizontal, is removed from the plate. Find the centre of mass of this plate from the lower left corner. Note the centre of mass about for the z-axis is neglected in the answer, since it is obviously at e/2.
- I found the center of mass for the square aluminum plate before and after a cylinder part of it was removed and subtracted the values but that didn't seem to work.
do MOMENTS not masses
M with center of mass at center, XM, YM, ZM
subtract m with center at xm, ym, zm
now have m' with center x', y', z'
m' = M-m
m x + m' x' = M XM
so
m' x' = M XM - m x
To find the center of mass of the square aluminum plate after the cylinder is removed, you need to consider the mass distribution of both the remaining square plate and the cylinder.
First, let's find the center of mass of the square aluminum plate without the cylinder:
1. Identify the coordinates of the lower left corner of the plate as (0, 0, 0).
2. Since the plate is square and has dimensions L = 4.8 m, the coordinates of the other corners are (L, 0, 0), (L, L, 0), and (0, L, 0).
3. The center of mass of the plate can be calculated by taking the average of these coordinates:
x-coordinate of center of mass = (0 + L + L + 0) / 4 = L / 2 = 4.8 / 2 = 2.4 m
y-coordinate of center of mass = (0 + 0 + L + L) / 4 = L / 2 = 4.8 / 2 = 2.4 m
z-coordinate of center of mass = 0 m (Since it is at the same height as the lower left corner.)
Therefore, the center of mass of the square aluminum plate, before the cylinder is removed, is located at (2.4 m, 2.4 m, 0 m) from the lower left corner.
Next, let's consider the cylinder that is removed from the plate:
1. The cylinder has a radius R = 1.5 m and is located at a distance d = 1.2 m from the center of the plate. The cylinder is at an angle of 20 degrees with the horizontal.
2. We can calculate the x and y coordinates of the center of the cylinder by using trigonometry:
x-coordinate of center of cylinder = d * cos(20 degrees) = 1.2 * cos(20°) = 1.1247 m
y-coordinate of center of cylinder = d * sin(20 degrees) = 1.2 * sin(20°) = 0.4115 m
3. Since the center of mass of the cylinder is at the same height as the lower left corner of the plate, the z-coordinate of the center of mass of the cylinder is 0 m.
Therefore, the center of mass of the cylinder, with respect to the lower left corner, is located at (1.1247 m, 0.4115 m, 0 m).
To find the center of mass of the square aluminum plate after the cylinder is removed, calculate the difference in the coordinates of the center of mass of the plate (found earlier) and the center of mass of the cylinder:
x-coordinate of center of mass of plate - x-coordinate of center of mass of cylinder = 2.4 m - 1.1247 m = 1.2753 m
y-coordinate of center of mass of plate - y-coordinate of center of mass of cylinder = 2.4 m - 0.4115 m = 1.9885 m
z-coordinate of center of mass of the plate remains the same, which is 0 m.
Therefore, the center of mass of the square aluminum plate, after the cylinder is removed, is located at (1.2753 m, 1.9885 m, 0 m) from the lower left corner.
Note: The z-coordinate of the center of mass does not change, as the removed cylinder is at the same height as the lower left corner of the plate.