- #1

CAF123

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## Homework Statement

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A circle of radius ##a##, with diameter ##AB##, is drawn on a sheet of paper which lies on a smooth horizontal table. The paper is pivoted with a pin at ##A## and has moment of inertia ##4ma^2## about a vertical axis through ##A##. An insect of mass ##m## walks around the circle with uniform speed ##2a\Omega## relative to the paper. The angle ##q## turned through by the paper, measured from an initial instant at which the insect passes through ##B## and at which the paper is momentarily at rest, is taken as a generalised co-ordinate; ##p## denotes the corresponding generalised momentum. Show that the lagrangian function for the system is

$$L = 2ma^2 [(1 + \cos^2 \Omega t)\dot{q}^2 + 2\Omega \dot{q} \cos^2 \Omega t]$$

2. Homework Equations

2. Homework Equations

L = T - V

## The Attempt at a Solution

My set up is shown in a picture below. ##q## is measured from ##AB## to the vertical axis and ##\Omega## is measured from ##AB## to the line connecting the centre of the circle to the mass ##m##. With this, $$x = a \cos \left(\frac{\pi}{2} - q\right) + a \cos\left(\frac{\pi}{2} - (q-\Omega)\right)$$ and similar expression for ##y## are the coordinates of the beetle in this frame. I just wanted to check if my set up is fine and that I have defined the angles ##q## and ##\Omega## correctly as given in the question.