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Energy - An interesting one-dimensional system is the simple pendulum, consisting of a point mass m, fixed to the end of a massless rod (length l)

An interesting one-dimensional system is the simple pendulum, consisting of a point mass m, fixed to the end of a massless rod (length l), whose other end is pivoted from the ceiling to let it swing freely in a vertical plane, as shown in Figure 4.26. The pendulum's position can be specified by its angle σ from the equilibrium position. (It could equally be specified by its distance s from equilibrium-indeed s=lσ-but the angle is a little more convenient.) (a)Prove that the pendulum's potential energy (measured from the equilibrium level) is

U(σ)=mgL_little(l-cosσ)

Write down the total energy E as a function of σ_dot. (b) Show that by differentiation your expression for E with respect to t you can get the equation of motion for σ and that the equation of motion is just the familiar Ґ=Ialpha (Where Ґ is the torque, I is the moment of inertia, and alpha is the angular acceleration σ_double-dot). (c) Assuming that the angle σ remains small throughout the motion, solve for σ(t) and show that the motion is periodic with period
τ_0=2π√(L_little/g)

I need help with this problem. Outline of solution please

Solution Summary

An interesting one-dimensional system is the simple pendulum, consisting of a point mass m, fixed to the end of a massless rod (length l).

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