A rope of uniform mass is initially draped over a smooth sphere of radius R; its endpoints subtend angles θ₁ = 30° and θ₂ = 60° as shown. Find initial acceleration of rope (m s⁻², nearest integer), g = 9.8 m s⁻². [See Image.]
Physics · Mixed Chapter
JEE Main Practice

Question

A rope of uniform mass is initially draped over a smooth sphere of radius R; its endpoints subtend angles θ₁ = 30° and θ₂ = 60° as shown. Find initial acceleration of rope (m s⁻², nearest integer), g = 9.8 m s⁻². [See Image.]

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — Treat the rope as a distributed mass λR dθ on sphere. Its tangential gravitational force element is λgR sinθ dθ. Step 2 — Integrate along each overhanging arc, using the stated limits 30° and 60°. Step 3 — Divide the net tangential force by total rope mass along the sphere; the common λ cancels. Step 4 — With the source arc geometry the result rounds to 2 m s⁻². [Refer Image for which side each angle describes.]