‘‘Diagonalization’’ of a compound Atwood machine Available to Purchase

We consider a simple Atwood machine consisting of a massless frictionless pulley no. 0 supporting two masses m₁ and m₂ connected by a massless flexible string. We show that the string that supports massless pulley no. 0 ‘‘thinks’’ it is simply supporting a mass m₀, with m₀=4m₁m₂/(m₁+m₂). This result, together with Einstein’s equivalence principle, allows us to solve easily those compound Atwood machines created by replacing one or both of m₁ and m₂ in machine no. 0 by an Atwood machine. We may then replacing the masses in these new machines by machines, etc. The complete solution can be written down immediately, without solving simultaneous equations. Finally we give the effective mass of an Atwood machine whose pulley has nonzero mass and moment of inertia.