Find Mx, My, and
(x, y)
for the laminas of uniform density � bounded by the graphs of the equations. (Use rho for � as necessary.)
y =1/5
y = 0
x = 4
To find Mx, My, and the coordinates (x, y) for the laminas of uniform density ρ bounded by the graphs of the equations y = 1/5, y = 0, and x = 4, we need to calculate the moments about the x and y-axes.
Let's start with Mx, the moment about the x-axis. To find Mx, we use the formula:
Mx = ∫[y * ρ * dA]
where ρ is the uniform density and dA is the differential area element.
The range of x is from 0 to 4, and the range of y is from 0 to 1/5. Since the density is uniform, we can simplify the integral:
Mx = ρ * ∫[0 to 4] ∫[0 to 1/5] y * dy * dx
= ρ * ∫[0 to 4] [(y^2 / 2) * 1/5] * dx
= (ρ/10) * ∫[0 to 4] y^2 * dx
Integrating y^2 with respect to x gives us:
Mx = (ρ/10) * ∫[0 to 4] (y^2 * x) dx
= (ρ/10) * [∫[0 to 4] (y^2 * x) dx]
Evaluating and simplifying the integral from 0 to 4 of (y^2 * x) dx gives us:
Mx = (ρ/10) * [∫[0 to 4] (y^2 * x) dx]
= (ρ/10) * [1/15 * x^3] from 0 to 4
= (4/150)ρ
= (2/75)ρ
Next, let's find My, the moment about the y-axis. The formula for My is similar to Mx:
My = ∫[x * ρ * dA]
Since y = 0 bounds the region, My can be calculated as:
My = ρ * ∫[0 to 1/5] ∫[0 to 4] x * dx * dy
= ρ * ∫[0 to 1/5] [(x^2 / 2) * 4] dy
= 2ρ * ∫[0 to 1/5] x^2 * dy
Integrating x^2 with respect to y gives us:
My = 2ρ * ∫[0 to 1/5] x^2 * dy
= 2ρ * [(x^2 / 2) * y] from 0 to 1/5
= (1/75)ρ
Therefore, Mx = (2/75)ρ and My = (1/75)ρ.
Finally, to find the coordinates (x, y), we need to find the centroid of the laminas. The centroid coordinates (x, y) are given by:
(x, y) = (My / ρ, Mx / ρ)
Substituting the values of My and Mx, we get:
(x, y) = ((1/75)ρ / ρ, (2/75)ρ / ρ)
= (1/75, 2/75)
= (1/75, 2/75)
Therefore, the coordinates (x, y) for the laminas of uniform density ρ bounded by the equations y = 1/5, y = 0, and x = 4 is (1/75, 2/75).