Question

A fluid is in solid body rotation in a cylindrical container of radius $R$ rotating with an angular velocity $\Omega = (0, 0, \Omega)$. The circulation per unit area around a circular loop in the horizontal plane of radius $r$ ($r<R$), whose center coincides with the axis of rotation is

• A. $r \Omega$
• B. $r \Omega^2$
• C. $r^2 \Omega$
• D. $r^2 \Omega^2$

Hint:

In solid body rotation, the velocity at any point is proportional to its distance from the axis of rotation. The circulation follows from this relationship.

Answer 1

The Correct Option is A

In solid body rotation, the velocity at any point is proportional to the distance from the axis of rotation, i.e., $v = r\Omega$. The circulation per unit area, which is the line integral of velocity around a closed loop, is given by:
\[ \Gamma = \oint \vec{v} \cdot d\vec{r}. \] For a circular loop of radius $r$, the circulation becomes:
\[ \Gamma = r \Omega, \] since the velocity is tangential and is given by $v = r \Omega$. Thus, the circulation per unit area is proportional to $r \Omega$.
Thus, the correct answer is:
\[ \boxed{\text{(A) } r\Omega}. \]