Assume you are given a sample of material and asked to determine its age. You the measure isotopes in the rock and determine that it has 125 mg of Radium-228 (the parent isotope) and 875 mg of Actinium-229 (the daughter isotope). How old is the rock sample if Radium-228 has a half life of 5.75 years?

Consider a double atwood machine constructed as follows: a mass 4m is suspended from a string that passes over a massless pulley on frictionless bearings. the other end of this string supports a second similar pulley, over which passes a second string supporting a mass of 3m at one end and m at the other. using two suitable generalized coordinates, set up the lagrangian and use the lagrange equations to find the acceleration of the mass 4m when the system is released. explain why the top pulley rotates even though it carries equal weights on each side.