A uniform rod of mass m and length l can rotate freely on a smooth horizontal plane about a vertical axis hinged at point H . A point mass having the same mass m coming with an initial speed u perpendicular to the rod, strikes the rod and sticks to it at a distance of (3 l/4) from hinge point. The angular velocity of the rod just after collision is (36 u/n l) . Find the value of n .

Question

A uniform rod of mass m and length l can rotate freely on a smooth horizontal plane about a vertical axis hinged at point H . A point mass having the same mass m coming with an initial speed u perpendicular to the rod, strikes the rod and sticks to it at a distance of (3 l/4) from hinge point. The angular velocity of the rod just after collision is (36 u/n l) . Find the value of n . <img class="img-fluid question-image" alt="Question" src="https://cdn.tardigrade.in/q/nta/p-eigpxrw8i0iuniak.jpg" /> (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Angular Momentum conservation about hinge, Li=Lf ⇒ mu((3 l/4))=((m l2/3) + m ((3 l/4))2)ω ⇒ u((3/4))=ω l× (43/48) ⇒ ω =(36 u/43 l)

Angular Momentum conservation about hinge, Li​=Lf​ ⇒mu(43l​)=(3ml2​+m(43l​)2)ω⇒u(43​)=ωl×4843​ ⇒ω=43l36u​

Angular Momentum conservation about hinge,
$L_{i} = L_{f}$
$\Rightarrow m u (\frac{3 l}{4}) = (\frac{m l ^{2}}{3} + m (\frac{3 l}{4})^{2}) ω \Rightarrow u (\frac{3}{4}) = ω l \times \frac{43}{48}$
$\Rightarrow ω = \frac{36 u}{43 l}$