By Erwin Schrödinger

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11 The potential energy U (x) between two atoms in a diatomic molecule can be expressed approximately by U (x) = − a b + 12 x6 x where x is the separation of the atoms and a and b are constants. (a) Obtain an expression for the force between the two atoms and hence show that the equilibrium separation xo of the atoms is equal to (2b/a)1/6 . (b) Show that the system will oscillate √ with SHM when slightly displaced from equilibrium with a frequency equal to k/m, 4/3 where m is the reduced mass and k = 36a(a/2b) .

So long as the amplitude of oscillation falls within the range where the two curves coincide the pendulum will execute SHM. 11 The solid curve represents the potential energy U of a simple pendulum as a function of its angular displacement θ . The dotted line represents the potential energy U (θ ) of a simple harmonic oscillator for which the potential energy is proportional to θ 2 . For small angular amplitudes, where the two curves overlap, a simple pendulum behaves as a simple harmonic oscillator.

If An occurs at time to then An = x(to ) = A0 exp(−γ to /2) cos ωto and An+1 = x(to + T ) = A0 exp[−γ (to + T )/2] cos ω(to + T ). 2 A graph of x = A0 exp(−γ t/2) cos ωt illustrating the decay in amplitude of a damped harmonic oscillator. 8), which forms an envelope of the oscillations. Since cos ωto = cos ω(to + T ) we have An γT = exp An+1 2 . 9) We see that successive maxima decrease by the same fractional amount. e. ln An An+1 = γT , 2 is called the logarithmic decrement and is a measure of this decrease.

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