Engineering Mechanics: Statics and Dynamics, 14th Edition
Authors: Russell C. Hibbeler
ISBN-13: 978-0133915426
See our solution for Question 61P from Chapter 17 from Hibbeler's Engineering Mechanics.
Problem 61P
Step-by-Step Solution
We are given the following data:
The value of horizontal force is ${\bf{P}} = 100\;{\rm{N}}$
The mass of reel is $m = 300\;{\rm{kg}}$.
The radius of gyration is ${k_O} = 0.6\;{\rm{m}}$.
The value of radius $r$ is $r = 0.75\;{\rm{m}}$.
The value of angle $\theta $ is $\theta = 20^\circ $.
We are asked to determine the initial angular acceleration.
Step 2
We will draw a free body diagram of the reel.
Here, $W$ is the weight of the reel and ${N_A}$ and ${N_B}$ are the normal reaction force at point $A$ and $B$.
Step 3
The formula to calculate the mass moment of inertia of the reel about $O$ is given by,
\[{I_O} = m{\left( {{k_O}} \right)^2}\]Substitute all the known values in the above formula.
\[\begin{array}{c} {I_O} = \left( {300\;{\rm{kg}}} \right){\left( {0.6\;{\rm{m}}} \right)^2}\\ = 108\;{\rm{kg}} \cdot {{\rm{m}}^{\rm{2}}} \end{array}\]Step 4
The formula to calculate the value of angular acceleration of the reel is given by,
\[\begin{array}{c} \sum {{M_O}} = {I_O}\alpha \\ - \left[ {\left( {\bf{P}} \right)\left( r \right)} \right] = \left[ {\left( {{I_O}} \right)\left( { - \alpha } \right)} \right] \end{array}\]Substitute all the known values in the above equation.
\[\begin{array}{c} - \left[ {\left( {100\;{\rm{N}}} \right)\left( {0.75\;{\rm{m}}} \right)} \right] = \left[ {\left( {108\;{\rm{kg}} \cdot {{\rm{m}}^{\rm{2}}}} \right)\left( { - \alpha } \right)} \right]\\ \alpha \approx 0.694\;{\rm{rad/}}{{\rm{s}}^{\rm{2}}} \end{array}\]