Question

A bead constrained to move in a tube of diameter $10\text{cm}$ is initially placed at the bottommost point of the tube. What minimum speed should be given to the bead so that it just reaches the topmost point in the tube?
Take $g=10{\text{m/s}}^2$

• A. $2\text{m/s}$
• B. $1\text{m/s}$
• C. $\sqrt{5}\text{m/s}$
• D. $\sqrt{2}\text{m/s}$

Answer 1

The correct option is D $\sqrt{2}\text{m/s}$

At the top most point of the tube, bead will have zero velocity i.e $v=0$ for just reaching at B

Applying energy conservation from point $A$ to point $B$,

$P{E}_A+K{E}_A=P{E}_B+K{E}_B$
Taking reference for $PE=0$ at point $A$

$\Rightarrow 0+\frac{1}{2}m{u}^2=mg\left(2R\right)+\frac{1}{2}m(0{)}^2$

$\Rightarrow u^2-4gR=0$

$\therefore u=\sqrt{4gR}$
Since, diameter $D=10\text{cm}$
Radius, $r=\frac{10}{2}\text{cm}=5\times {10}^{-2}\text{m}$
$\Rightarrow u=\sqrt{4\times 10\times 5\times {10}^{-2}}$

$\therefore u=u_{min}=\sqrt{2}\text{m/s}$
This is required minimum speed for bead to reach at the top of tube.

Hence option D is the correct answer