A thin uniform rod of length 2m, cross-sectional area 'A' and density 'd' is rotated about an axis passing through the centre and perpendicular to its length with angular velocity . If the value of in terms of its rotational kinetic energy E is then the

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Solution

Q.
A thin uniform rod of length 2m, cross-sectional area 'A' and density 'd' is rotated about an axis passing through the centre and perpendicular to its length with angular velocity $ω$ . If the value of $ω$ in terms of its rotational kinetic energy E is $\sqrt{\frac{α E}{A d}}$ then the value of $α$ is ________ . [2023]

Ans.
(3)

${(KE)}_{rotational} = \frac{1}{2} I ω^{2} = E$

$E = \frac{1}{2} \frac{m l^{2}}{12} ω^{2} = \frac{1}{2} \frac{d A l^{3}}{12} ω^{2}$

$E = \frac{d A {(2)}^{3}}{24} ω^{2}$

$\sqrt{\frac{3 E}{d A}} = ω \Rightarrow α = 3$

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