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14 Figure 1: Sliding pendulum 4. (38pt) Practice power expansion and initial condition A block of mass M is free to slide on a horizontal bar without any friction. A mass m is attached to the bottom of the block with a massless rod of length 1 and can oscillate freely in the same plane as the horizontal bar. (a) (5pt) Write down the Lagrangian of the system (b) (7pt) Find the equations of motion. (c) (10pt) Assume that the pendulum oscillation is constrained to small angles around zero. Write the equation of motion in that case. (d) (6pt) Find the frequency of the small oscillations of the system. (e) (10pt) Given the initial conditions of θ(t=0-60 and 6(t=0)=6yB(M m) (A), find 6(t) as a function of the g, M. m, l and θ。. (f) Given that x(t 0) = To find the equation that describes the movement of the bob in the horizontal direction
Please be as detailed as possible cause I want to understand it. Thanks!

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Answer #1

Start with the cartesian coordinates, (X, Y, Z) and (x, y, z), measured from a single inertial frame. The 4 constraints can b

The Euler-Lagange equations are (M + m)X+mm, cosdd=constant =

X------->>> x

Apply Lagranges equations: and

We also have a second constant of the motion, the energy rcosconstant For small displacements we have where we have kept no t

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