Engineering Mechanics: Statics and Dynamics, 14th Edition
Authors: Russell C. Hibbeler
ISBN-13: 978-0133915426
See our solution for Question 33FP from Chapter 12 from Hibbeler's Engineering Mechanics.
Problem 33FP
Step-by-Step Solution
We are given the speed of the car as $v = 55\,{\rm{ft/s}}$ and the radius of the turn as $r = 400\,{\rm{ft}}$.
We are asked to determine the angular velocity $\dot \theta $ of the radial line $OA$ at this instant.
Step 2
Find the derivative of $r$ using the following relation.
\[\dot r = \frac{d}{{d\theta }}r\]On substituting the known values in the above equation we get,
\[\begin{array}{c} \dot r = \frac{d}{{d\theta }}\left( {400\,{\rm{ft}}} \right)\\ = 0 \end{array}\]Step 3
Find the radial component of velocity of the car using the following relation.
\[{v_r} = \dot r\]On substituting the known values in the above equation we get,
\[{v_r} = 0\]Step 4
Find the transverse component of velocity of the car using the following relation.
\[{v_\theta } = r\dot \theta \]On substituting the known values in the above equation we get,
\[{v_\theta } = \left( {400\,{\rm{ft}}} \right)\dot \theta \,......\left( 1 \right)\]Step 5
Find the angular velocity $\dot \theta $ using the following relation.
\[v = \sqrt {v_r^2 + v_\theta ^2} \]On substituting the known value of equation (1) in the above equation we get,
\[v = \sqrt {v_r^2 + {{\left[ {\left( {400\,{\rm{ft}}} \right)\dot \theta } \right]}^2}} \]On substituting the known values in the above equation we get,
\[\begin{array}{c} 55\,{\rm{ft/s}} = \sqrt {0 + {{\left[ {\left( {400\,{\rm{ft}}} \right)\dot \theta } \right]}^2}} \\ 3025\,{\rm{f}}{{\rm{t}}^2}{\rm{/}}{{\rm{s}}^2} = {\left[ {\left( {400\,{\rm{ft}}} \right)\dot \theta } \right]^2}\\ {{\dot \theta }^2} = \frac{{3025\,{\rm{f}}{{\rm{t}}^{\rm{2}}}{\rm{/}}{{\rm{s}}^{\rm{2}}}}}{{160000\,{\rm{f}}{{\rm{t}}^2}}}\\ \dot \theta = 0.1375\,{\rm{rad/s}} \end{array}\]