A circular uniform disk of radius R has a hole of radius r (r < R/2). The centre of the hole is R/2 from the centre of the disk. Find the position of the centre of mass. Surface density of the disk is σ.
To find the position of the center of mass of the circular disk with a hole, we can divide the disk into two regions: the outer disk with radius R and the inner disk with radius r.
First, let's find the position of the center of mass of the outer disk. Since the disk has uniform density, we can consider it as an equivalent point mass located at its center. The mass of the outer disk can be calculated as the difference between the mass of the full disk (πR²σ) and the mass of the inner disk (πr²σ).
The mass of the outer disk is given by:
M_outer = πR²σ - πr²σ
= σ(πR² - πr²)
= πσ(R² - r²)
Next, we need to find the position of the center of mass of the outer disk. Since the hole is symmetrically placed with respect to the center of the disk, the center of mass of the outer disk will also coincide with the center of the full disk, which is at a distance of R/2 from the center of the hole.
Therefore, the position of the center of mass of the outer disk is (R/2, 0).
Now, let's find the position of the center of mass of the inner disk. Since the hole is centered at R/2 from the center of the full disk, the center of mass of the inner disk will be located at the center of the hole, which is at a distance R/2 from the center of the full disk.
Therefore, the position of the center of mass of the inner disk is (R/2, 0).
To find the position of the center of mass of the composite object (disk with a hole), we can use the concept of the center of mass as the weighted average of the positions of individual components.
Let's denote the position of the center of mass of the composite object as (x_cm, y_cm).
Since the masses of the outer and inner disks are the same (πσ(R² - r²)), the x-coordinate of the center of mass of the composite object will be the average of the x-coordinates of the centers of mass of the outer and inner disks:
x_cm = (R/2 + R/2)/2
= R/2
Since the y-coordinate of the center of mass of the outer and inner disks is 0, the y-coordinate of the center of mass of the composite object will also be 0:
y_cm = 0
Therefore, the position of the center of mass of the composite object (disk with a hole) is (R/2, 0).