A pendulum made of point mass m on a massless rod of length is hanging...
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A pendulum made of point mass m on a massless rod of length is hanging from a mass M sliding freely without friction or external forcing along a horizontal rail Write the Lagrangian of the system How many degrees of freedom does it have Solve Euler Lagrange equations and write down the nonlinear governing equations of motion Show that one of the variables is ignorable cyclic and that this leads to an integral of motion which expresses the conservation of linear momentum of the system along the rail Assume that at time zero the pendulum was slightly deflected away from vertical but nothing was moving Write the equation of infinitesimally small oscillations of the pendulum where the deflection and its rate of change are so small that powers higher than 1 of these quantities can all be neglected What is the frequency of small oscillations how different is it from the standard pendulum Discuss the limits of m M 0 m M
A pendulum made of point mass m on a massless rod of length is hanging from a mass M sliding freely without friction or external forcing along a horizontal rail Write the Lagrangian of the system How many degrees of freedom does it have Solve Euler Lagrange equations and write down the nonlinear governing equations of motion Show that one of the variables is ignorable cyclic and that this leads to an integral of motion which expresses the conservation of linear momentum of the system along the rail Assume that at time zero the pendulum was slightly deflected away from vertical but nothing was moving Write the equation of infinitesimally small oscillations of the pendulum where the deflection and its rate of change are so small that powers higher than 1 of these quantities can all be neglected What is the frequency of small oscillations how different is it from the standard pendulum Discuss the limits of m M 0 m M
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