A uniform rod of length 12 metre and mass m is hinged at midpoint and lies horizontally at rest. Rod is free to rotate in vertical plane about hinge O. A particle of mass m is released from height h as shown in figure. Particle collides with end A and sticks to it. Calculate minimum height h (in meter) so that system can complete vertical circular motion.
Physics · Mixed Chapter
JEE Mains

Question

A uniform rod of length 12 metre and mass m is hinged at midpoint and lies horizontally at rest. Rod is free to rotate in vertical plane about hinge O. A particle of mass m is released from height h as shown in figure. Particle collides with end A and sticks to it. Calculate minimum height h (in meter) so that system can complete vertical circular motion.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: The particle strikes at distance r=6 m from hinge with speed u=√(2gh). Angular momentum about the hinge is conserved during the sticking collision. Step 2: Rod inertia about midpoint is m(12)²/12=12m; stuck particle contributes m(6)²=36m. Thus total I=48m. Step 3: m u×6=48m ω, hence ω=u/8. Immediately after collision the rotational energy is ½(48m)(u²/64)=3mu²/8. Step 4: During rotation the rod's centre of mass remains at hinge, while the attached particle must climb 6 m to the uppermos…