Find the z-component of the location of the mass center of the body shown in figurewhich is a hemisphere glued to the solid cylinder of the same density. Assume L = 1.9Rand R= 1.4 m.RFigure is from "Engineering Mechanic An Introduction to Dynamics", McGill and King.

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Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Find the z-component of the location of the mass center of the body shown in figure
which is a hemisphere glued to the solid cylinder of the same density. Assume L = 1.9
Rand R= 1.4 m.
R
Figure is from "Engineering Mechanic An Introduction to Dynamics", McGill and King.

Expert Solution

Step 1

The z component of the location of the mass of the center body is

For the cylindrical section

$m_{1} = ρ \times (π R^{2} L)$

For the semicircular section

$m_{2} = ρ \times (\frac{2}{3} π R^{3})$

The z component of the location of the mass of the center body is

$\begin{array}{rcl} Z & = & \frac{\frac{m_{1} L}{2} + m_{2} \times (L + \frac{3 R}{8})}{m_{1} + m_{2}} \\ = & \frac{\frac{(ρ π R^{2} \times 1.9 R) (1.9 R)}{2} + (ρ \times \frac{2}{3} π R^{3}) \times (1.9 R + \frac{3 R}{8})}{(ρ π R^{2} \times 1.9 R) + (ρ \times \frac{2}{3} π R^{3})} \\ = & \frac{11.34 R^{4} ρ + 4.76 R^{4} ρ}{5.96 R^{3} ρ + 2.09 R^{3} ρ} \\ = & \frac{R^{4} ρ [11.34 + 4.76]}{R^{3} ρ [5.96 + 2.09]} \\ = & 2 R \\ = & 2 (1.4 m) \\ = & 2.8 m \end{array}$

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