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Grade: upto college level
        A sphere of mass 20 kg is suspended by a metal wire of unstretched length 4 m and diameter 1 mm. When in equilibrium, there is a clear gap of 2 mm between the sphere and the floor. The sphere is gently pushed aside so that the wire makes an angle θ with the vertical and is released. Find the maximum value of θ so that the sphere does not rub the floor. Young modulus of the metal of the wire is 2.0 × 1011 N m-2. Make appropriate approximations. 
5 years ago

Answers : (1)

Aditi Chauhan
askIITians Faculty
396 Points
							Sol. . At equilibrium ⇒ T = mg
When it moves to an angle θ, and released, the tension the T’ at lowest point is
⇒ T’ = mg + mv2/r
The change in tension is due to centrifugal force  ∆T = mv2/r ….(1)
⇒ Again, by work energy principle,
⇒ 1/2mv2 – 0 = mgr(1 - cosθ)
⇒ v2 = 2gr (1- cosθ)
So, ∆T = m[2gr(1 – cos θ)]/r = 2mg(1 – cos θ)
⇒ F = ∆T
⇒ F = YA ∆L/L = 2mg – 2mg cos θ ⇒ 2mg cos θ = 2mg - YA ∆L/L
=  cos θ = 1 - YA ∆L/L(2mg)

5 years ago
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