use formula (A1X1 + A2X2)/A1+A2 where X is distance from origin of both since com of half disc is 4r/3pie put the values you will get answer .hope you will be satisfied

The problem asks us to find the center of mass of a uniform plate with semicircular inner and outer boundaries of radii R1 and R2. To solve this problem, we will break it down into several steps.

Step 1: Identify the Geometry
We are given a uniform plate with semicircular inner and outer boundaries. Let's consider the plate to be lying in the xy-plane. The center of the plate can be taken as the origin, and the x-axis and y-axis can be chosen accordingly.

Step 2: Determine the Mass Distribution
Since the plate is uniform, the mass is distributed evenly throughout the plate. We can assume that the mass per unit area is constant.

Step 3: Divide the Plate into Smaller Elements
To simplify the problem, we can divide the plate into smaller elements. Let's consider an infinitesimally small element at position (x, y) on the plate, with an area dA.

Step 4: Express the Mass of Each Element
The mass of an infinitesimally small element can be expressed as dm = ρdA, where ρ is the mass per unit area.

Step 5: Express the Position Vector of Each Element
The position vector of each element can be expressed as r = xi + yj, where i and j are the unit vectors in the x and y directions, respectively.

Step 6: Determine the Center of Mass
To find the center of mass, we need to calculate the coordinates (X, Y) of the centroid. The coordinates can be obtained using the following equations:

X = (1/M) ∫xdm
Y = (1/M) ∫ydm

Where M is the total mass of the plate.

Step 7: Evaluate the Integrals
To evaluate the integrals, we need to express dm in terms of the variables x and y. Since we have already expressed dm as ρdA, we can substitute dA with the appropriate expression for the area element. For a semicircle, the area element can be expressed as dA = Rdθ, where R is the radius of the semicircle and θ is the angle.

Step 8: Simplify and Solve
By substituting the expressions for dm and dA, we can simplify the integrals and solve for X and Y. The values obtained will give us the coordinates of the center of mass.

In summary, to find the center of mass of a uniform plate with semicircular inner and outer boundaries, we need to divide the plate into smaller elements, express the mass and position vector of each element, evaluate the integrals, and solve for the coordinates of the center of mass.