Question

A thin bar of mass 'M' and length 'L' is free to rotate about a fixed horizontal axis through a point at its end.The bar is brought to a horizontal position and then released. The angular velocity when it reaches the lowest point is:

• A. directly proportional to its length and inversely proportional to its mass.
• B. independent of mass and inversely proportional to the square root of its length
• C. dependent only upon the acceleration due to gravity.
• D. directly proportional to its length and inversely proportional to the acceleration due to gravity.

Answer 1

The correct option is B independent of mass and inversely proportional to the square root of its length

Here $\omega =\sqrt{\frac{6g}{L}}\sin \frac{\theta }{2}$

Where, $\theta =$ angle of rod when rotated

At the most lower end

$\Rightarrow \theta =180°$

$\therefore \omega =\sqrt{\frac{2g}{L}}\sin 90°$

$=\sqrt{\frac{2g}{L}}$

$\therefore \omega \alpha \frac{1}{\sqrt{L}}$

Thus, $\omega$ is independent of mass and inversely proportional to $\sqrt{L}.$

Hence, the answer is independent of mass and inversely proportional to the square root of its length.