Question

Please be as detailed as possible cause I want to understand it. Thanks!

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

Answer 1

Start with the cartesian coordinates (X, Y, Z) and (x, y, z), measured form a single inertial frame. The 4 constraints can be written Y = Z z = 0 and (x - X)^2 + y^2 - r^2 = 0. This means we have two generalized coordinates (duh, we just said that!) In terms of Cartesian coordinates, T = 1/2 M X^2 + 1/2 m (x^2 + y^2) ---(1) Now write transformation equations from the Cartesian coordinates to the generalized coordinates. X = X + r sin theta y = -r cos theta X = X Hence abovedot x = abovedot X + r abovedot theta cos theta abovedot y = r abovedot theta sin theta abovedor X = abovedot X and using these in Equations 1 T = 1/2 M X^2 + 1/2 m [avovedot X^2 + (r abovedot theta)^2 + 2 abovedot X r abovedot theta cos theta] V = -mgr cos theta L = 1/2 M abovedot X^2 + 1/2 m [abovedot X^2 + (r abovedot theta)^2 + 2 abovedot X^2 r abovedot theta cos theta] + mgr cos theta..

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