A bead of mass m is in unstable equilibrium at top of a smooth circular wire frame which is fixed. The bead is slightly disturbed and slowly brought to equilibrium position where spring is along tange

Question

A bead of mass m is in unstable equilibrium at top of a smooth circular wire frame which is fixed. The bead is slightly disturbed and slowly brought to equilibrium position where spring is along tangent to the circular wire frame. If the spring is initially at natural length, what is spring constant k (in N/m) ? Take $R = 3.75 m, l_{0} = 6 m, m = 1.3 kg$ .

$R = 3.75 m, l_{0} = 6 m, m = 1.3 kg .$

TestHub · Accepted Answer

Geometry of the tangent position: ( l 0 ​ + x ) 2 + R 2 = ( l 0 ​ + R ) 2 , so x 2 + 2 l 0 ​ x − 2 l 0 ​ R = 0 With l 0 ​ = 6 m and R = 3.75 m: x = 2 l 0 ​ ​ = 3 m Tangential equilibrium: k x = m g sin θ , where sin θ = l 0 ​ + R l 0 ​ + x ​ = 9.75 9 ​ = 13 12 ​ k = 3 1.3 × 10 × 12/13 ​ = 4 N/m

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