A metal wire of unstretched length 4 m and diameter 1 mm suspends a sphere of mass 20 kg . At equilibrium, a clear gap of 2 mm is present between the sphere and the floor. The sphere is gently pulled aside until the wire forms an angle with the vertical, and is then released. Determine the maximum value of such that the sphere avoids rubbing the floor. The Young modulus of the wire's metal is 2.0 10¹¹ N/m ^2 . Make a

Step-by-step solution

Let the original unstretched length of the wire be L . At equilibrium, the tension in the wire is T₀ = mg . The elongation of the wire at equilibrium is: L₀ = mgL AY When the sphere passes through the lowest point (mean position) after being released from an angle , its velocity v is given by conservation of energy: v^2 = 2gL(1 - ) The tension T at the lowest point is: T = mg + mv^2 L = mg + 2mg(1 - ) For the sphere to just avoid rubbing the floor, the maximum elongation of the wire at the lowest point must be L =