Discussion
A fluid is in solid body rotation in a cylindrical container of radius \(R\) rotating with an angular velocity \(\boldsymbol{\Omega} = (0, 0, \Omega)\). The circulation per unit area around a circular loop in the horizontal plane of radius \(r\) (\(r \lt R\)), whose center coincides with the axis of rotation is
Answer: A
The circulation per unit area is \(2\Omega\).
In solid body rotation the speed at radius r is \(v = \Omega r\). Around the circle of radius r:
\[ \begin{aligned} \Gamma &= \oint \mathbf{v}\cdot d\mathbf{l} = \Omega r \cdot 2\pi r = 2\pi\Omega r^{2} \\ \frac{\Gamma}{\pi r^{2}} &= 2\Omega \end{aligned} \]
This is the vorticity of solid body rotation, twice the angular velocity (Stokes' theorem). \(\Omega/2\) and \(\Omega/4\) divide instead of multiply, and \(\Omega^{2}\) has the wrong dimensions for a vorticity.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development