Question

A metal bead of mass $500\text{g}$ placed inside a hollow circular tube of radius $10\text{m}$ is initially at rest at the bottom-most point. It is given an initial velocity of $15\text{m/s}$ such that it moves along the hollow circular tube. Can the metal bead subtend an angle of ${30}^{\circ }$ with the positive horizontal axis? If yes, what is the kinetic energy at this point? [Take $g=10{\text{m/s}}^2$]

• A. The kinetic energy at this point is $56.25\text{J}$
• B. It cannot subtend an angle of ${30}^{\circ }$ with the positive horizontal axis.
• C. The kinetic energy at this point is $18.75\text{J}$
• D. The kinetic energy at this point is $75\text{J}$

Answer 1

The correct option is B It cannot subtend an angle of ${30}^{\circ }$ with the positive horizontal axis.


Mass of bead $=500\text{g}=0.5\text{kg}$
From the figure, we know that when the metal bead subtends an angle of ${30}^{\circ }$ at the centre, then,
$\sin {30}^{\circ }=\frac{h}{r}=\frac{h}{10}$
$\Rightarrow h=10\times \sin {30}^{\circ }=10\times \frac{1}{2}=5\text{m}$

Initial $KE$ of the ball
$=\frac{1}{2}m{v}^2=\frac{1}{2}m\times {15}^2$
$=\frac{1}{2}\times 0.5\times 225=56.25\text{J}$

If the bead can just reach the point where it subtends angle ${30}^{\circ }$ with the horizontal axis, its $KE$ at that point will be zero
Potential energy of the bead at the point {with respect to bottom-most point}
$=mg(r+h)=0.5\times 10(10+5)$
$=75\text{J}$
Thus, we can see that the total energy of the body has increased which violates the law of conservation of energy.
Hence the metal bead cannot reach the point where it subtends an angle of ${30}^{\circ }$ with the positive horizontal axis.