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Chapter 16: Problem 3

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

Short Answer

Expert verified
Answer: To find the center of mass of a thin plate with variable density, follow these steps: 1. Define the density function, 蟻(x, y), which represents the mass per unit area at each point on the thin plate. 2. Establish a coordinate system on the thin plate and define a small area element, dA = dx dy, at a position (x, y). 3. Calculate the mass of the area element, dM, by multiplying the density function by the area element: dM = 蟻(x, y) dA = 蟻(x, y) dx dy. 4. Integrate the mass function over the entire area of the thin plate to find the total mass, M: M = 鈭埆 dM = 鈭埆 蟻(x, y) dx dy. 5. Calculate the moments of the mass distribution about the x and y axes: Mx = 鈭埆 y蟻(x, y) dx dy, and My = 鈭埆 x蟻(x, y) dx dy. 6. Determine the x and y coordinates of the center of mass by dividing the moments of the mass distribution by the total mass: x_c = My/M and y_c = Mx/M.

Step by step solution

01

Define the density function

Define the density function, 蟻(x, y), which represents the mass per unit area at each point on the thin plate. The function will depend on the given conditions of the problem, and it can be either a scalar constant if the density is uniform or a function of x and y if the density varies over the plate.
02

Set up coordinate system and define area element

Establish a coordinate system on the thin plate, usually with the origin at the corner or center of the plate. Then, define a small area element, dA = dx dy, that represents a small rectangular portion of the thin plate at a position (x, y).
03

Determine the mass of the area element

Calculate the mass of the area element, dM, by multiplying the density function by the area element: dM = 蟻(x, y) dA = 蟻(x, y) dx dy.
04

Calculate the total mass of the thin plate

Integrate the mass function over the entire area of the thin plate to find the total mass, M: M = \int \int dM = \int \int 蟻(x, y) dx dy.
05

Find the moments of the mass distribution

Calculate the moments of the mass distribution about the x and y axes. The moment about the x-axis is found by integrating the product of the mass function and the y coordinate over the entire area: Mx = \int \int y蟻(x, y) dx dy. Similarly, the moment about the y-axis is found by integrating the product of the mass function and the x coordinate over the entire area: My = \int \int x蟻(x, y) dx dy.
06

Calculate the coordinates of the center of mass

Finally, determine the x and y coordinates of the center of mass by dividing the moments of the mass distribution by the total mass: x_c = My/M and y_c = Mx/M.

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