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 tried finding the center of masses for the square aluminum plate and the cylinder removed from the plate, and then subtracted the cylinder's center of mass from that of the square plate. Apparently I'm not getting the right answer though so any help would be appreciated.
To find the center of mass of the aluminum plate with the cylinder removed, we need to consider the center of mass of both components separately and then combine them.
First, let's find the center of mass of the square aluminum plate. Given that the plate has a length L = 4.8 m and width e = 0.9 m, the center of mass of the plate about the lower left corner can be calculated as follows:
The x-coordinate of the center of mass is:
x_plate = L/2 = 4.8/2 = 2.4 m
The y-coordinate of the center of mass is:
y_plate = e/2 = 0.9/2 = 0.45 m
So the center of mass of the square aluminum plate is located at (2.4 m, 0.45 m) from the lower left corner.
Next, let's find the center of mass of the cylinder. The cylinder's center of mass is at a distance of d = 1.2 m from the center of the plate and at an angle of 20 degrees with the horizontal. To calculate its coordinates, we can use trigonometry:
The horizontal displacement of the cylinder's center of mass is:
x_cylinder = d * cos(20 degrees) = 1.2 * cos(20 degrees) ≈ 1.117 m
The vertical displacement of the cylinder's center of mass is:
y_cylinder = d * sin(20 degrees) = 1.2 * sin(20 degrees) ≈ 0.407 m
So the center of mass of the cylinder is located at (1.117 m, 0.407 m) from the lower left corner.
Finally, to find the center of mass of the plate with the cylinder removed, we subtract the coordinates of the cylinder's center of mass from the plate's center of mass:
x_cm = x_plate - x_cylinder = 2.4 - 1.117 ≈ 1.283 m
y_cm = y_plate - y_cylinder = 0.45 - 0.407 ≈ 0.043 m
Therefore, the center of mass of the plate with the cylinder removed from the lower left corner is approximately located at (1.283 m, 0.043 m).