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Unformatted text preview: CHAPTER 2. OSCILLATIONS 2.1. THE HARMONIC OSCILLATOR Consider the pendulum shown. The small object, mass m , at the end of the pendulum, is moving on a circle of radius L , so the component of its velocity tangential to the circle is L ˙ θ Hence its tangential acceleration is L ¨ θ and so by * F = M * a we have mL ¨ θ = mg sin θ. An obvious solution is θ = 0. This is called an EQUI 1 LIBRIUM solution, meaning that θ is a CONSTANT function. This means that if you set θ = 0 initially, then θ will remain at 0 and the pendulum will not move — which of course we know is correct. There is ANOTHER equilibrium solution, θ = π . Again, IN THEORY, if you set the pendulum EXACTLY at θ = π , then it will remain in that position for ever. IN REALITY, of course, it won’t! Because the slightest puff of air will knock it over! So this equilibrium is very different from the one at θ = 0. This is a very important distinction! Equilibrium is said to be STABLE if a SMALL push away from equilibrium REMAINS small. If the small 2 push tends to grow large, then the equilibrium is UNSTABLE. Obviously this is important for engi neers! Especially you want vibrations of structures, engines, etc to remain small. Let’s look at θ = π . By Taylor’s theorem, near θ = π , we have f ( θ ) = f ( π ) + f ( π )( θ π ) + 1 2 f 00 ( π )( θ π ) 2 + ... Now sin( π ) = 0, sin ( π ) = cos( π ) = 1, sin 00 ( π ) = sin( π ) = 0 etc so sin( θ ) = 0 ( θ π ) 0 + 1 6 ( θ π ) 3 etc For small deviations away from π , θ π is small, 3 ( θ π ) 3 is much smaller, etc, so we can approximate sin( θ ) ≈  ( θ π ) so our equation is approximately ML ¨ θ = mg sin θ = mg ( θ π ) . Let φ = θ π , so ¨ φ = ¨ θ , and now ¨ φ = g L φ. The general solution is φ = Ae ( √ g/L ) t + Be ( √ g/L ) t so θ = φ + π = Ae ( √ g/L ) t + Be ( √ g/L ) t + π. 4 As you know, the exponential function grows very quickly; so even if θ is close to π initially, it won’t stay near to it very long! Very soon, θ will arrive either at θ = 0 or 2 π , far away from θ = π . The equilibrium is UNSTABLE! How long does it take for things to get out of control? That is determined by p g/L or rather p L/g , which has units of TIME. Note that it takes longer to fall over if L is large. SUMMARY : The equation ¨ φ = + g L φ is a symptom of INSTABILITY. The system is at equilibrium, but it will run away uncontrollably on a time scale fixed by p L/g . 5 Now what about θ = 0? Here of course we use Tay lor’s theorem around zero, f ( θ ) = f (0) + f (0) θ + 1 2 f 00 (0) θ 2 + ... sin( θ ) = 0 + θ 1 6 θ 3 + ... so sin( θ ) ≈ θ and we have approximately mL ¨ θ = mgθ or ¨ θ = g L θ = ω 2 θ with ω 2 = g/L . That minus sign is crucial!...
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This note was uploaded on 03/29/2008 for the course MA 1506 taught by Professor Mcnines during the Spring '08 term at National University of Singapore.
 Spring '08
 McNines

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