Suppose that she pushes on the sphere tangent to its surface with a steady force of f = 75 n and that the pressured water provides a frictionless support. how long will it take her to rotate the sphere one time, starting from rest?

The time taken to rotate the sphere one time is,  t = 22 s

Explanation:

Given data,

The mass of the sphere, m = 8200 kg

The radius of the sphere, r = 90 cm

                                             = .9 m

The force applied by the girl, F = 75 N

The moment of inertia of the sphere is,

                            I = 2/5 mr²

                              = (2/5) 8200 x (.9)²

                              = 2657 kg·m²

The torque,

                            τ = I α

                             75 x 0.9 = 2657 x α

                              α = 0.0254 rad/s²

The angular displacement,

                            θ = ½αt²

                             2π =  ½ x 0.0254 rad/s² x t²

                                t = 22 s

Hence, the time taken to rotate the sphere one time is,  t = 22 s