Engineering Mechanics: Statics and Dynamics, 14th Edition
Authors: Russell C. Hibbeler
ISBN-13: 978-0133915426
See our solution for Question 13P from Chapter 17 from Hibbeler's Engineering Mechanics.
Problem 13P
Step-by-Step Solution
We are given a wheel in form of a thin ring having four spokes made from slender rods.
The mass of the thin ring is, ${m_r} = 10{\rm{ kg}}$.
The number of spokes in the ring is, $n = 4$.
The mass of each spoke is, ${m_s} = 2{\rm{ kg}}$.
The radius of the wheel is, $r = 500{\rm{ mm}}$.
We are asked to determine the wheel’s moment of inertia about an axis perpendicular to the page and passing through point $A$.
Step 2
The diagrammatic representation of the wheel is given below:
Step 3
The moment of inertia of the thin ring having four spokes about an axis perpendicular to the page and passing through point $A$ is given by,
\[\begin{array}{c} {I_o} = 4 \times \left( {\frac{{{m_s}{d^2}}}{{12}}} \right) + {m_r}{r^2}\\ {I_o} = 4 \times \left( {\frac{{2\;{\rm{kg}} \times {{\left( {1\;{\rm{m}}} \right)}^2}}}{{12}}} \right) + \left( {10\;{\rm{kg}}} \right) \times {\left( {{\rm{0}}{\rm{.5}}\;{\rm{m}}} \right)^2}\\ {I_o} = 3.167\;{\rm{kg}} \cdot {{\rm{m}}^2} \end{array}\]Step 4
The moment of inertia of the whole wheel about an axis perpendicular to the page and passing through point $A$ is given by,
\[\begin{array}{c} {I_A} = {I_o} + \left( {{m_r} + n{m_s}} \right){r^2}\\ {I_A} = \left( {3.167\;{\rm{kg}} \cdot {{\rm{m}}^2}} \right) + \left[ {\left( {10\;{\rm{kg}} + \left( 4 \right) \times 2\;{\rm{kg}}} \right)} \right] \times {\left( {0.5\;{\rm{m}}} \right)^2}\\ {I_A} = \left( {3.167\;{\rm{kg}} \cdot {{\rm{m}}^2}} \right) + \left( {4.5\;{\rm{kg}} \cdot {{\rm{m}}^2}} \right)\\ {I_A} = 7.67\;{\rm{kg}} \cdot {{\rm{m}}^2} \end{array}\]