Find Mx, My, and (x, y) for the lamina of uniform density ρ bounded by
y=1/2x ,y>0 x=2
This is discussed clearly at
http://www.ltcconline.net/greenl/courses/202/multipleIntegration/MassMoments.htm
Where do you get stuck?
To find the values of Mx, My, and (x, y) for the given lamina, we need to understand the concepts of Moments of Inertia and Center of Mass.
1. Moment of Inertia (Ix and Iy):
The moment of inertia measures the resistance of an object to changes in rotation. In the context of a lamina, we calculate the moment of inertia with respect to the x-axis (Ix) and the y-axis (Iy).
2. Center of Mass (x̄, ȳ):
The center of mass is the point at which the total mass of an object is considered to be concentrated. For a lamina, the center of mass is denoted by (x̄, ȳ) and is calculated separately for both the x and y coordinates.
Now, let's calculate Mx, My, and (x, y) step by step:
Step 1: Calculation of Ix:
For a lamina with uniform density bound by y = 1/2x and x = 2, the formula to calculate Ix is:
Ix = ∫(y² * dm)
To calculate the integral, we need to express dm in terms of a differential area element.
dm = ρ * dA
The differential area element dA is obtained by considering the rectangular differential strip formed by y = 1/2x, y = 0, and x = 2. The length of the strip is dy, and the width is dx.
Therefore, the differential area element can be expressed as:
dA = dx * dy
Substituting this into the expression for dm, we get:
dm = ρ * dx * dy
So, the integral expression for Ix becomes:
Ix = ∫(y² * ρ * dx * dy)
Step 2: Calculation of My:
Similarly, the formula to calculate My is:
My = ∫(x² * dm)
Using the same approach as Step 1 to express dm and substituting into the equation for My, we obtain:
My = ∫(x² * ρ * dx * dy)
Step 3: Calculation of (x, y):
The center of mass coordinates (x̄, ȳ) are calculated using the following formulas:
x̄ = My / M
ȳ = Mx / M
where M is the total mass of the lamina.
Step 4: Calculation of M:
The total mass M is given by the equation:
M = ∫(dm)
Using the same approach as Step 1, substituting the expression for dm, we have:
M = ∫(ρ * dx * dy)
Now, you can compute the values of Mx, My, and (x, y) by performing the integrals and calculating the necessary quantities.