Question

A circular tube of radius ‘R’ and cross-sectional radius ‘r’ (r << R) is filled completely with iron balls of radius ‘r’. Iron balls are just fitting into the tubes. The tension in the tube when it is rotated about its axis perpendicular to its plane with angular velocity ‘w’ –

• A. $\frac{4}{3}$prw²r³R
• B. $\frac{4}{3}\pi\rho\omega^{2}r^{2}R^{2}$
• C. $\frac{2}{3}\pi\rho\omega^{2}r^{3}R$
• D. $\frac{2}{3}\pi\rho\omega^{2}r^{2}R^{2}$

Answer 1

Answer: (D)

$\frac{2}{3}\pi\rho\omega^{2}r^{2}R^{2}$

Answer 2

Q r < < R, mass/length can be given by

l = $\frac{m}{2r}$= $\frac{\frac{4}{3}\pi r^{3}\rho}{2r}$.$\frac{2}{3}\pi r^{2}\rho$

Now,

Tension in tube is given by

T = (l . w²R)R = $\frac{2}{3}\pi r^{2}\rho$. w².R.R

= $\frac{2}{3}\pi\rho\omega^{2}r^{2}R^{2}$ Q