Question

2) In the above diagram of a pendulum, a mass, m, hangs on a massless cord of length, L, from the ceiling. Suppose the mass is pulled to the side so that the string makes an angle θ with the vertical. It is then released from rest at time t-0 and so it swings back and forth with no friction of any kind. You must use the energy method as used in the first problem (mass on a spring) in the most recent homework. Take the zero of potential energy to be at the lowest point of the swing (ie, when θ =0) a) Obtain the potential and kinetic energies in terms of mL g.9,9 and d9/dt and hence set up the integral equation that would give θ as a function of t but DO NOT TRY TO INTEGRATE-it is messy. Make sure the integrals are in terms of the right variables with the right limits etc. For small o the answer to part (a) can be approximated by judicious use of Taylor series and integrated to b) get θ as a function of t. Use this result to find the time taken to fall from the (small) θ° to the lowest point (9-0) and confirm that that time is independent of θ0.

Answer 1

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