A hinged equilateral wire frame of mass 3m and side ℓ oscillates in a vertical plane under gravity and two massless springs of constants k₁ and k₂ as shown. Determine the integer coefficient in the displayed small-oscillation frequency. [See Image.]
Physics · Mixed Chapter
JEE Main Practice

Question

A hinged equilateral wire frame of mass 3m and side ℓ oscillates in a vertical plane under gravity and two massless springs of constants k₁ and k₂ as shown. Determine the integer coefficient in the displayed small-oscillation frequency. [See Image.]

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — The three uniform sides together have mass 3m, and their moment of inertia about hinge A is found by adding each segment’s inertia about A. Step 2 — Determine gravitational restoring torque from the wire-frame centre of mass. Step 3 — Each spring contributes an incremental torque proportional to its change of length: τₛ = −(k₁a₁² + k₂a₂²)θ for small θ, with lever arms a₁ and a₂ from the diagram. Step 4 — The small-oscillation frequency is ω = √(κₑff/Iₐ), and f = ω/(2π). The source prin…