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A pendulum with a moving support point

A pendulum with a moving support pointGabriela Gonz alezSeptember 12, 2006 Consider a pendulum with massmhanging from a rod of lengthl. The support pointmoves horizontally with a known functionR(t) =X(t) i+Y(t) j. We can use the angle between the vertical and the pendulum rod as a generalized coordinate, the only oneneeded to describe the position vector of the massmisr=R+l(sin i+ cos j)The velocity isv= R+l (cos i sin j)The kinetic energy isT=12mv2=12m(| R|2+l2 2+ 2l ( Xcos Ysin )The potential energy isV= mg r= mgcos mgYand although time dependent (throughY(t), it is not dissipative, since it doesn t contain .The Lagrangian isL=T VL( , ,t) =12ml2 2+ml ( X(t) cos Y(t) sin ) +12m( X(t)2+ Y(t)2) +mglcos +mgY(t)Lagrange s equation is0 =ddt L L =ddt(ml2 +ml( Xcos Ysin )) (ml ( Xsin Ycos ) mglsin )=ml2 +ml( Xcos Ysin ) +mglsin =ml2 +ml(g Y) sin +ml Xcos 1We see that the equation of motion is unchanged if R= 0: this is because the support -would be moving with constant velocity, and thus it is just like setting up the system inanother inertial frame.))

• Vertical periodic motion: Y = Y 0 cos(Ωt), X = 0. 2Y 0 cosΩt)sinθ. The motion of the support adds an oscillating component to the gravitational acceleration. This is not a driven oscillator like the previous case, because the oscillatory driving force is vertical but the natural oscillatory pendulum motion is horizontal. If Y 0Ω2 ˝ g,

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  Motion, Oscillatory

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