Essential University Physics: Volume 1 (3rd Edition)

Published by Pearson
ISBN 10: 0321993721
ISBN 13: 978-0-32199-372-4

Chapter 10 - Exercises and Problems - Page 187: 73



Work Step by Step

We know: $I=\int .5r^2 dm$ Thus, we need to find dm: $dm=\frac{M}{\frac{1}{3}\pi R^2 h}\times\pi r^2 dz$ This can be simplified to: $dm=\frac{3Mz^2}{h^3}$ We plug this and the equation $r^2 = \frac{R^2z^2}{h^2}$ into the above integral to find: $I=\int_0^h .5 \frac{R^2z^2}{h^2} \cdot \frac{3Mz^2}{h^3}$ Factoring the constants out of the integral leaves: $ I=\frac{R^2}{2h^2} \cdot \frac{3M}{h^3} \int_0^h z^4 $ $I=\frac{R^2}{2h^2} \cdot \frac{3M}{h^3} \frac{h^5}{5}$ $I=\frac{3MR^2}{10}$
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