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Solution: Mass and center of mass Let S be a surface that
Chapter 11, Problem 69E(choose chapter or problem)
Let S be a surface that represents a thin shell with density p. The moments about the coordinate planes (see Section 13.6) are \(M_{y z}=\iint_{S} x \rho(x, y, z) d S\), \(M_{x z}=\iint_{S} y \rho(x, y, z) d S, M_{x y}=\iint_{S} z \rho(x, y, z) d S\).
The coordinates of the center of mass of the shell are \(\bar{x}=\frac{M_{y z}}{m}\), \(\bar{y}=\frac{M_{x z}}{m}, \bar{z}=\frac{M_{x y}}{m}\), where m is the mass of the shell. Find the mass and center of mass of the following shells. Use symmetry whenever possible.
The cylinder \(x^{2}+y^{2}=a^{2}, 0 \leq z \leq 2\), with density \(\rho(x, y, z)=1+z\)
Questions & Answers
QUESTION:
Let S be a surface that represents a thin shell with density p. The moments about the coordinate planes (see Section 13.6) are \(M_{y z}=\iint_{S} x \rho(x, y, z) d S\), \(M_{x z}=\iint_{S} y \rho(x, y, z) d S, M_{x y}=\iint_{S} z \rho(x, y, z) d S\).
The coordinates of the center of mass of the shell are \(\bar{x}=\frac{M_{y z}}{m}\), \(\bar{y}=\frac{M_{x z}}{m}, \bar{z}=\frac{M_{x y}}{m}\), where m is the mass of the shell. Find the mass and center of mass of the following shells. Use symmetry whenever possible.
The cylinder \(x^{2}+y^{2}=a^{2}, 0 \leq z \leq 2\), with density \(\rho(x, y, z)=1+z\)
ANSWER:Solution 69EStep 1 Consider the following cylindrical shell: The density is given by the function The coordinates of the center of mass of such a shell is given by formula: Where m is the mass of the shell and is defined as below: