Q. In the adjacent figure a uniform rod of length $L$ and mass $m$ is kept at rest in horizontal position on an elevated edge. The value of $x$ (consider the figure-3.119) is such that the rod will have maximum angular acceleration $\alpha$ as soon as it set free.

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A

$x$ is equal to $\frac{L}{2 \sqrt{3}}$

B

$\alpha$ is equal to $\frac{g \sqrt{3}}{2 L}$

C

$\alpha$ is equal to $\frac{g \sqrt{3}}{L}$

D

$x$ is equal to $\frac{L}{\sqrt{3}}$

Solution:

Apply $\tau=I \alpha$ about $O$
$m g x=\left(\frac{m L^{2}}{12}+m x^{2}\right) \alpha$

For $\alpha$ to be maximum $d \alpha / d x=0$
which gives $x=\frac{L}{2 \sqrt{3}}$ and $\alpha=\frac{\sqrt{3} g}{L}$

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Q. In the adjacent figure a uniform rod of length $L$ and mass $m$ is kept at rest in horizontal position on an elevated edge. The value of $x$ (consider the figure-3.119) is such that the rod will have maximum angular acceleration $\alpha$ as soon as it set free.

$x$ is equal to $\frac{L}{2 \sqrt{3}}$

$\alpha$ is equal to $\frac{g \sqrt{3}}{2 L}$

$\alpha$ is equal to $\frac{g \sqrt{3}}{L}$

$x$ is equal to $\frac{L}{\sqrt{3}}$

Apply $\tau=I \alpha$ about $O$ $m g x=\left(\frac{m L^{2}}{12}+m x^{2}\right) \alpha$ For $\alpha$ to be maximum $d \alpha / d x=0$ which gives $x=\frac{L}{2 \sqrt{3}}$ and $\alpha=\frac{\sqrt{3} g}{L}$

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1. The work done by a particle moving with a velocity of $0.7\, c$ (where c is the velocity of light) in empty space free of electromagnetic field and far away from all matter is:

2. The machine gun fires $240 \, $ bullets per minute. If the mass of each bullet is $10 \, g$ and the velocity of the bullets is $600 \, m \, s^{- 1}$ ,the power (in $kW$ )of the gun is

4. A particle of mass $m$ is moving in a circular path of constant radius $r$ such that its centripetal acceleration $a_{c}$ is varying with time $t$ as $a_{c}=k^{2} \, rt^{2}$ , where $k$ is a constant. The power delivered to the particle by the force acting on it is

5. A ball of mass $1\, kg$ collides with a wall with speed $8\, ms ^{-1}$ and rebounds on the same line with the same speed. If mass of the wall is taken as infinite, the work done by the ball on the wall is

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5. A barometer is constructed using a liquid (density $= 760 \,kg/m^{3})$ What would be the height of the liquid column, when a mercury barometer reads $76\, cm$ ? (density of mercury $= 13600 kg/m^{3})$

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Apply $\tau=I \alpha$ about $O$
$m g x=\left(\frac{m L^{2}}{12}+m x^{2}\right) \alpha$

For $\alpha$ to be maximum $d \alpha / d x=0$
which gives $x=\frac{L}{2 \sqrt{3}}$ and $\alpha=\frac{\sqrt{3} g}{L}$

5. A barometer is constructed using a liquid (density $= 760 \,kg/m^{3})$ What would be the height of the liquid column, when a mercury barometer reads $76\, cm$ ?
(density of mercury $= 13600 kg/m^{3})$