Question

Consider the vector field \(\vec{V}\) consisting of the velocities of points on a thin horizontal disc of radius \(R = 2 \, \text{m}\), moving anticlockwise with uniform angular speed \(\omega = 2 \, \text{rad/sec}\) about an axis passing through its center. If \(V = |\vec{V}|\), then which of the following options is/are CORRECT? (In the options, \(\hat{r}\) and \(\hat{\theta}\) are unit vectors corresponding to the plane polar coordinates \(r\) and \(\theta\)). \includegraphics[width=0.75\linewidth]{image57.png}

• A. \(\nabla . \vec{V} = 2 \hat{r}\)
• B. \(\nabla . \vec{V} = 2\)
• C. \(\vec{\nabla} \times \vec{V} = 4 \hat{Z}\), where \(\hat{Z}\) is a unit vector perpendicular to the \((r, \theta)\) plane
• D. \(\nabla^2 \vec{V} = \frac{4}{3} \text{ at } r = 1.5 \, \text{m}\)

Hint:

To calculate divergence, curl, and Laplacian in cylindrical coordinates, use the appropriate formulas for vector fields in polar coordinates. Pay special attention to the symmetry of the problem.

Answer 1

The Correct Option is A, C, D

- The given vector field describes a rotating disc with velocity \(V = r\omega\) where \(\omega = 2\) rad/sec.
- Using the standard expressions for divergence and curl in cylindrical coordinates, we find the following results: - \(\nabla . \vec{V} = 2 \hat{r}\), which corresponds to option (A).
- \(\vec{\nabla} \times \vec{V} = 4 \hat{Z}\), matching option (C).
- The Laplacian \(\nabla^2 \vec{V}\) evaluated at \(r = 1.5 \, \text{m}\) gives \(\frac{4}{3}\), as stated in option (D).