A semi-circular plate of radius 'r' has density ρ=ρ0r where'r' is the distance from the center.The position of the centre of mass from the center 'o' is 1.5yR/2π where 'y' is(Integer Type)

To find the center of mass of a semi-circular plate with a density that varies with the distance from the center, we need to take a systematic approach. The density function given is ρ = ρ₀r, where ρ₀ is a constant and r is the distance from the center. This means that the density increases linearly as we move away from the center of the plate. Let's break down the steps to find the center of mass.

Understanding the Geometry

First, we need to visualize the semi-circular plate. It occupies the upper half of a circle with radius R. The center of the circle is at the origin (0,0) in a Cartesian coordinate system. The plate extends from -R to R along the x-axis and from 0 to R along the y-axis.

Setting Up the Coordinates

In polar coordinates, we can express the position of any point on the semi-circle as:

• x = r cos(θ)
• y = r sin(θ)

where θ ranges from 0 to π for the semi-circle. The area element in polar coordinates is given by:

dA = r dr dθ

Calculating the Mass

The total mass M of the semi-circular plate can be calculated by integrating the density over the area:

M = ∫∫ ρ dA = ∫(0 to π) ∫(0 to R) (ρ₀r) (r dr dθ)

Substituting the density function, we have:

M = ρ₀ ∫(0 to π) ∫(0 to R) r² dr dθ

Calculating the inner integral:

∫(0 to R) r² dr = [r³/3] from 0 to R = R³/3

Now, substituting this back, we get:

M = ρ₀ ∫(0 to π) (R³/3) dθ = ρ₀ (R³/3) [θ] from 0 to π = ρ₀ (R³/3) π

Finding the Center of Mass

The center of mass (x̄, ȳ) can be found using the formulas:

• x̄ = (1/M) ∫∫ x ρ dA
• ȳ = (1/M) ∫∫ y ρ dA

Since the plate is symmetric about the y-axis, we can conclude that x̄ = 0. Now, let's calculate ȳ:

ȳ = (1/M) ∫(0 to π) ∫(0 to R) (r sin(θ)) (ρ₀r) (r dr dθ)

Substituting the density function:

ȳ = (1/M) ∫(0 to π) ∫(0 to R) (ρ₀r² sin(θ)) (r dr dθ)

Now, we can factor out ρ₀:

ȳ = (ρ₀/M) ∫(0 to π) sin(θ) dθ ∫(0 to R) r³ dr

Calculating the inner integral:

∫(0 to R) r³ dr = [r⁴/4] from 0 to R = R⁴/4

Now, the integral of sin(θ) from 0 to π is 2:

Thus, we have:

ȳ = (ρ₀/M) (2) (R⁴/4)

Substituting M = ρ₀ (R³/3) π:

ȳ = (ρ₀ (2) (R⁴/4)) / (ρ₀ (R³/3) π) = (2R/4) * (3/π) = (3R/2π)

Final Result

Therefore, the position of the center of mass from the center 'o' is given by:

ȳ = (3R/2π)

In your question, you mentioned the center of mass as 1.5yR/2π, which can be interpreted as a specific case where y is an integer. This means that for different integer values of y, the center of mass will shift accordingly, but the formula remains consistent with the derived expression. This illustrates how the center of mass is influenced by the distribution of mass in the plate.