Appreciated if helped - igcse based question

Aplayground merry-go-round has a radius of 4.6 m and a moment of inertia of 200 kg-m2 and turns with negligible friction about a vertical axle through its center. a child applies a 26.0 n force tangentially to the edge of the merry-go-round for 15.0 seconds. if the merry-go-round is initially at rest, how much work did the child do on the merry-go-round?

Asolid cylinder and a cylindrical shell have the same mass, same radius, and turn on frictionless, horizontal axles. the cylindrical shell has light-weight spokes connecting the shell to the axle. a rope is wrapped around each cylinder and tied to blocks of equal masses that are held the same height above the ground. both blocks are released simultaneously. the ropes do not slip. which block hits the ground first? or is it a tie?

Suppose that f : rn → rm and that a ∈ k, where k is a connected subset of rn . suppose further that for each x ∈ k there exists a δx > 0 such that f(x) = f(y) for all y ∈ bδx (x). prove that f is constant on k; that is, f(x) = f(a) for all x ∈ k