A uniform rod of mass 'm' and length 'l' is kept vertical with the lower end cla

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A uniform rod of mass 'm' and length 'l' is kept vertical with the lower end clamped. It is slightly pushed to let it fall down under gravity. Find its angular speed when the rod is passing through its lowest position. Neglect any friction at the clamp. What will be the linear speed of the free end at this instant?

Options:

A . √6gl,√6gl

B . √2gl,√6gl

C . √6gl,√3gl

D . √3gl,√6gl

Answer: Option A
:
A

A Uniform Rod Of Mass 'm' And Length 'l' Is Kept Vertical Wi...

As the rod reaches its lowest position, the centre of mass is lowered by a distance l. Its gravitational potential energy is decreased by mgl. As no energy is lost against friction, this should be equal to the increase in the kinetic energy. As the rotation occurs about the horizontal axis through the clamped end, the moment of inertia is I =

m l^{2} / 3

. Thus,

\frac{1}{2} I ω^{2} = m g l

\frac{1}{2} (\frac{m l^{2}}{3}) ω^{2} = m g l

or

ω = \sqrt{\frac{6 g}{l}}

.
The linear speed of the free end is

V = I ω = \sqrt{6 g l}

.

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