A bead of mass 5 kg slides without friction on a fixed vertical ring of radius r=20 cm. A spring connects bead to top O, its natural length equals r. Initially ∠OCA=60°. At bottom B, ring normal reaction is zero. If k=100n N/m, find n (g=10 m/s²).
Physics · Mixed Chapter
JEE Main Practice

Question

A bead of mass 5 kg slides without friction on a fixed vertical ring of radius r=20 cm. A spring connects bead to top O, its natural length equals r. Initially ∠OCA=60°. At bottom B, ring normal reaction is zero. If k=100n N/m, find n (g=10 m/s²).

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: At A, spring length OA=2r sin(60°/2)=r; initially unstretched. At bottom B, OB=2r, spring extension=r. Step 2: Vertical drop from A to B is r(1+cos60°)=3r/2. Energy: mg(3r/2)=½mv²+½kr². Step 3: At B when normal reaction zero, radial balance upward: kr−mg=mv²/r (spring force upward, gravity downward). Step 4: Eliminate v²: 3mg r = mv²+kr²; with mv²=kr²−mgr ⇒3mgr=2kr²−mgr ⇒ k=2mg/r=2×5×10/0.2=500 N/m. Step 5: k=100n⇒n=5.