Question

Explain how to find the center of mass of a thin plate with a variable density.

Short answer

Answer: To find the center of mass of a thin plate with variable density, follow these essential steps: 1. Obtain the expression for the variable density, denoted as ρ(x, y). 2. Compute the total mass, M, of the plate by integrating the mass of small rectangular elements over the entire plate. 3. Calculate the moment of mass about the x and y axes, denoted as M_x and M_y, by integrating the product of the coordinates and density over the plate. 4. Determine the coordinates of the center of mass, (x̅, ȳ), using the formulas x̅ = M_x/M and ȳ = M_y/M.

Step-by-step solution

Expression for Density

Recall that the given density is variable, so we will represent it as \(\rho(x, y)\), where \((x, y)\) are the coordinates of a point on the plate. Given this density function, the next step is to find the mass \(dm\) of a small rectangular element with sides \(\Delta x\) and \(\Delta y\) located at the point \((x, y)\). We can find this mass by multiplying the area of the rectangle by the density at that point: $$dm = \rho(x, y) \cdot \Delta x \cdot \Delta y.$$

Total Mass of the Plate

To find the total mass of the plate, we need to integrate the mass of these small rectangular elements over the entire plate. Let's assume that the plate occupies a region \(R\) in the \(xy\)-plane. The total mass can be found by integrating over this region: $$M = \int\int_R \rho(x, y) \, dA = \int\int_R \rho(x, y) \, dx \, dy.$$ Depending on the specific function \(\rho(x, y)\), we will need to adjust the limits of integration accordingly.

Moment of Mass

To find the center of mass, we first need to find the moment of the mass about the \(x\) and \(y\) axes. The moment of the mass about the \(x\)-axis (\(M_y\)) and the \(y\)-axis (\(M_x\)) are found as follows: $$M_y = \int\int_R x\rho(x, y) \, dA = \int\int_R x\rho(x, y) \, dx \, dy,$$ $$M_x = \int\int_R y\rho(x, y) \, dA = \int\int_R y\rho(x, y) \, dx \, dy.$$ Again, we will need to adjust the limits of integration depending on the specific function \(\rho(x, y)\).

Center of Mass

Now that we have the total mass \(M\) and moments \(M_x\) and \(M_y\), we can find the coordinates of the center of mass, denoted as \((\overline{x}, \overline{y})\). The coordinates are given by these formulas: $$\overline{x} = \frac{M_x}{M}, \quad\text{and}\quad \overline{y} = \frac{M_y}{M}.$$ After calculating \(\overline{x}\) and \(\overline{y}\), we will have the center of mass of the thin plate with variable density.