A piece of uniform sheet metal measures 25 cm by 25 cm. If a circular piece with a radius of 5.0 cm is cut
from the center of the sheet, where is the sheet’s center of mass now?
Its been awhile I worked with radius so can you refresh my memory?
To find the new center of mass after a circular piece is cut from the center of the sheet, we need to determine the new mass distribution.
1. Calculate the mass of the sheet:
The density of the sheet metal is required to calculate the mass. If the density is not given, we can assume a typical value for sheet metal, such as 7.85 g/cm³.
The area of the sheet is (25 cm) x (25 cm) = 625 cm².
The mass of the sheet is the area multiplied by the density, so: mass = 625 cm² x 7.85 g/cm³ = 4906.25 g.
2. Calculate the mass of the circular piece:
The area of a circle is given by the formula A = πr², where r is the radius.
The area of the circular piece is therefore A = π(5.0 cm)² = 78.54 cm².
Assuming the same density as the sheet metal, the mass of the circular piece is: mass = 78.54 cm² x 7.85 g/cm³ = 616.08 g.
3. Determine the position of the new center of mass:
Since the circular piece is cut from the center, the center of mass of the sheet and the circle coincided before cutting.
After cutting, the center of mass will shift towards the remaining material.
We can consider the sheet as a rectangular shape with a center of mass at its geometric center, which is at (12.5 cm, 12.5 cm) in this case.
Since the mass distribution is not uniform within the sheet, the exact coordinates need to be recalculated using the mass of each section.
To find the x-coordinate of the new center of mass:
x = [(m1 * x1) + (m2 * x2)] / (m1 + m2)
where m1 and x1 are the mass and x-coordinate of one section, and m2 and x2 are the mass and x-coordinate of the other section.
The sheet can be divided into two sections: one with the circular cut-out, and the other without it.
Let's denote the section without the circular cut-out as section 1, and the section with the circular cut-out as section 2.
Section 1 has a mass of 4289.17 g (4906.25 g - 616.08 g) and a center of mass at (12.5 cm, 12.5 cm).
Section 2 has a mass of 616.08 g and a center of mass at the geometric center, which is (12.5 cm, 12.5 cm) as well.
Plugging in the values, we have:
x = [(4289.17 g * 12.5 cm) + (616.08 g * 12.5 cm)] / (4289.17 g + 616.08 g)
x = 12.5 cm
Therefore, the x-coordinate of the new center of mass is 12.5 cm.
The y-coordinate of the new center of mass remains the same as the original center of mass, as the mass is uniformly distributed along the y-axis.
Conclusion:
After cutting the circular piece from the center of the sheet, the new center of mass is at coordinates (12.5 cm, 12.5 cm).