A thin uniform rod of length L and certain mass is kept on a frictionless horizontal table with a massless string of length L fixed to one end (top view is shown in the figure). The other end of the string is pivoted to a point O . If a horizontal impulse P is imparted to the rod at a distance x=L /

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A thin uniform rod of length L and certain mass is kept on a frictionless horizontal table with a massless string of length L fixed to one end (top view is shown in the figure). The other end of the string is pivoted to a point O . If a horizontal impulse P is imparted to the rod at a distance x=L / n from the mid-point of the rod (see figure), then the rod and string revolve together around the point O , with the rod remaining aligned with the string. In such a case, the value of n is _______ .

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Correct answer: A. A

Linear impulse F d t= momentum aligned & = m ( V _ cm -0 ) \ & P = m ( r _ cm ) \ & = m ( L + L 2 ) aligned P = m ( 3 ~L 2 ) ...(i) Angular impulse dt = angular momentum aligned & r Fdt = L \ & aligned & r Fdt = I ( -0), and I is moment of inertia about axis of rotation. \ & aligned & ( L + L 2 + x ) P = ( I _ cm + md ^2 ) \ &= ( mL ^2 12 + m ( L + L 2 )^2 ) aligned \ & ( 3 ~L 2 + x ) P = mL ^2 ( 1 12 + ( 3 2 )^2 ) aligned aligned ( 3 L 2 +x ) P= mL ^2 ( 7 3 ) ...(ii) Divide eq.-(i) & (ii) aligned & ( 3 L 2 +x )= L ( 7 3 ) ( 3 2 ) \ & 3 L 2 +x=L ( 14 9 ) \ & x= L 18 aligned

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