66–69. Mass and center of mass Let S be a surface that represents athin shell with density p. The moments about the coordinate planes (seeSection 16.6) are My, = ls xp(x, y. z) dS, M = lsyp(x, y, z) dS,and M, = ls zp(x, y, z) dS. The coordinates of the center of mass ofM.*yzthe shell are I =mM,M,2r,and ī =where m is the mass ofmmthe shell. Find the mass and center of mass of the following shells. Usesymmetry whenever possible.hhThe constant-density half-cylinder x? + z? = a²,z 20V

Answered: 66–69. Mass and center of mass Let S be…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

66–69. Mass and center of mass Let S be a surface that represents a
thin shell with density p. The moments about the coordinate planes (see
Section 16.6) are My, = ls xp(x, y. z) dS, M = lsyp(x, y, z) dS,
and M, = ls zp(x, y, z) dS. The coordinates of the center of mass of
M.
*yz
the shell are I =
m
M,
M,
2r,
and ī =
where m is the mass of
m
m
the shell. Find the mass and center of mass of the following shells. Use
symmetry whenever possible.
h
h
The constant-density half-cylinder x? + z? = a²,
z 20
V

Expert Solution

Step 1

Center of mass is defined as the point at which whole mass distribution can be assumed to be concentrated. Center of mass of symmetrical structures of constant density lies on it's geometrical center.

Given:

$d e n s i t y : ρ s h a p e : x^{2} + z^{2} = a^{2}, - \frac{h}{2} < y < \frac{h}{2}, z \geq 0$

Step 2

According to given shape we can use this formula to compute moment about y-z plane :

$\begin{array}{rcl} M_{y z} & = & \iint x ρ d S \end{array}$

Defining cylindrical system:

$\begin{array}{rcl} x & = & a \cos ϕ \\ z & = & a \sin ϕ \\ y & = & y \\ s o, a & = & \sqrt{x^{2} + z^{2}} \\ s o, d S & = & a d ϕ d y \end{array}$

So integral becomes:

$\begin{array}{rcl} M_{y z} & = & \int_{- \frac{h}{2}}^{\frac{h}{2}} \int_{0}^{π} a \cos (ϕ) \cdot ρ \cdot a d ϕ d y \\ = & ρ a^{2} \int_{- \frac{h}{2}}^{\frac{h}{2}} d y \int_{0}^{π} \cos (ϕ) d ϕ \\ = & ρ a^{2} {[y]}_{- \frac{h}{2}}^{\frac{h}{2}} {[\sin ϕ]}_{0}^{π} \\ = & ρ a^{2} [\frac{h}{2} + \frac{h}{2}] [sinπ - \sin 0] \\ = & 0 \end{array}$

This moment about y-z plane is zero that verifies this shape is symmetrical about y-z plane.

Similarly for x-z plane:

$\begin{array}{rcl} M_{z x} & = & \iint y ρ d S \\ = & \int_{- \frac{h}{2}}^{\frac{h}{2}} \int_{0}^{π} y \cdot ρ \cdot a d ϕ d y \\ = & ρ a \int_{- \frac{h}{2}}^{\frac{h}{2}} y d y \int_{0}^{π} d ϕ \\ = & ρ a {[\frac{y^{2}}{2}]}_{- \frac{h}{2}}^{\frac{h}{2}} {[ϕ]}_{0}^{π} \\ = & \frac{ρ a}{2} [\frac{h^{2}}{4} - \frac{h^{2}}{4}] [π - 0] \\ = & 0 \end{array}$

Similarly for x-y plane:

$\begin{array}{rcl} M_{x y} & = & \iint z ρ d S \\ = & \int_{- \frac{h}{2}}^{\frac{h}{2}} \int_{0}^{π} a \sin (ϕ) \cdot ρ \cdot a d ϕ d y \\ = & ρ a^{2} \int_{- \frac{h}{2}}^{\frac{h}{2}} d y \int_{0}^{π} \sin (ϕ) d ϕ \\ = & ρ a^{2} {[y]}_{- \frac{h}{2}}^{\frac{h}{2}} {[- \cos ϕ]}_{0}^{π} \\ = & ρ a^{2} [\frac{h}{2} + \frac{h}{2}] [- \cos π + \cos 0] \\ = & ρ a^{2} h 2 \\ = & 2 ρ a^{2} h \end{array}$