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 parts: the outer disk (with radius R) and the hole (with radius r). We can then find the position of the center of mass for each part separately and use the principle of conservation of mass to determine the overall position of the center of mass.
Let's start by finding the position of the center of mass for the outer disk.
The mass of the outer disk can be calculated by multiplying the area of the disk by its surface density σ. The area of a disk of radius R is given by A = πR^2, so the mass of the outer disk is m1 = σ * A = σ * πR^2.
Now, let's find the position of the center of mass for the outer disk. The center of mass for a circular disk is located at its physical center, which is also the center of the circle. Therefore, the x-coordinate of the center of mass for the outer disk is x1 = 0, and the y-coordinate is y1 = 0.
Next, let's find the position of the center of mass for the hole.
Since the hole is a circular shape as well, we can use the same formula to find its mass. The area of a circle with radius r is given by A = πr^2, so the mass of the hole is m2 = σ * A = σ * πr^2.
Unlike the outer disk, the hole is not centered at the origin of the coordinate system. According to the problem statement, the center of the hole is R/2 from the center of the disk. This means that the x-coordinate of the center of mass for the hole is x2 = R/2, and the y-coordinate is y2 = 0.
To find the overall position of the center of mass, we need to consider the masses and positions of both the outer disk and the hole. The position of the center of mass, denoted as (x_cm, y_cm), is given by:
x_cm = (m1 * x1 + m2 * x2) / (m1 + m2)
y_cm = (m1 * y1 + m2 * y2) / (m1 + m2)
Substituting the values we found earlier, we have:
x_cm = (σ * πR^2 * 0 + σ * πr^2 * R/2) / (σ * πR^2 + σ * πr^2)
y_cm = (σ * πR^2 * 0 + σ * πr^2 * 0) / (σ * πR^2 + σ * πr^2)
Simplifying the expressions, we get:
x_cm = σ * r^2 * R / (2R^2 + r^2)
y_cm = 0
Therefore, the position of the center of mass is (x_cm, y_cm) = (σ * r^2 * R / (2R^2 + r^2), 0).