Engineering Mechanics: Statics and Dynamics, 14th Edition
Authors: Russell C. Hibbeler
ISBN-13: 978-0133915426
See our solution for Question 1FP from Chapter 16 from Hibbeler's Engineering Mechanics.
Problem 1FP
Step-by-Step Solution
We are given the angular position of gear as $\theta = 20\,{\rm{rev}}$ and the angular velocity as $\omega = 30\,{\rm{rad/s}}$.
We are asked to determine the constant angular acceleration and the time required.
Step 2
First convert the angular velocity in terms of radians using the following relation.
\[{\theta _r} = \theta \times \left( {\frac{{2\pi \,{\rm{rad}}}}{{1\,{\rm{rev}}}}} \right)\]On substituting the known value in the above equation we get,
\[\begin{array}{c} {\theta _r} = \left( {20\,{\rm{rev}}} \right) \times \left( {\frac{{2\pi \,{\rm{rad}}}}{{1\,{\rm{rev}}}}} \right)\\ = 40\pi \,{\rm{rad}} \end{array}\]Step 3
As the gear is initially at rest, the initial angular velocity is zero $\left( {{\omega _0} = 0} \right)$.
Find the angular acceleration using the following relation.
\[{\omega ^2} = \omega _0^2 + 2\alpha {\theta _r}\]On substituting the known value of equation (1) in the above equation we get,
\[\begin{array}{c} {\left( {30\,{\rm{rad/s}}} \right)^2} = 0 + 2\alpha \left( {40\pi \,{\rm{rad}}} \right)\\ \alpha = \frac{{{{\left( {30\,{\rm{rad/s}}} \right)}^2}}}{{\left( {80\pi \,{\rm{rad}}} \right)}}\\ \alpha = 3.58\,{\rm{rad/}}{{\rm{s}}^2} \end{array}\]Step 4
Find the time required by using the following relation.
\[\omega = {\omega _0} + \alpha t\]On substituting the known value in the above equation we get,
\[\begin{array}{c} \left( {30\,{\rm{rad/s}}} \right) = 0 + \left( {3.58\,{\rm{rad/}}{{\rm{s}}^{\rm{2}}}} \right)t\\ t = 8.38\,{\rm{s}} \end{array}\]