Question

1) Consider a block of mass M connected through the massless rigid rod to the massless circular track of radius a on a frictionless horizontal table (see the Figure). A particle of mass m is constrained to move on the vertical circular track. The distance between the center of the circular track and the center of mass of the block of mass M is constant and equal to L. Assume that there is no friction between the track and the particle. M (a) Let (x,0) be the coordinate of center of the circular track. Using 0 as one of the generalized coordinates, write the Lagrangian of the system. (b) Find the equations of motion. (c) Assume that 0 and are both small. Solve 0 as a function of time t and find the frequency of small oscillations.

Answer 1

We can write the lagrangian in terms of generalised cordinates x and theta. After writing the lagrangian the Equation of motion we can write in 2 coordinates.

To find the frequency of oscillations we have to write the kinetic and potential energy matrices assuming the small oscillations. Then by applying the secular equation, frequency can be found. Please find the attached solution.
→ TE NI 3m [li+acoso) + (psins oy") + 2 Mia m Pape-z T= M- 3m2 x² + a²8² copo + qže a core + a²0² sino] + 1 mx² (3 TE m 22 x² + a²0² + 2xé a coro + // Mx² → Total potential Energy of system. V= VitV₂ V = mgy, to V= - mgacose Lagrangian. T-V 5 L £m i++a+62+2 xacope) + + ļ mx² + mga core L Gen. Coordinates x,0.

Papez d dt OL ax OL ox mė a cose + M 11 → mät ma ö cose + mao (sing it Mä=o mät Mät mal ocoro-o? sino] o d OL de la OL Do d dt ma²6 + maxco maicono) - ( mai ė (-sino) mag sinejo ma²ö + maä coso max sino mario'sino + magsino = + ma² + macoso X żė (masino-masing) + magsino =o mało" + macoso že + magsino = 0

G for small oscillations. Page 3 K.E.) system TE Im (32+ a?o?+ 2a żó cosa) + Ź Mä? Small Oscillations sino o coso (Approximations) I- TE 5m (i+ Qo2 + 2a to (1 - 07:))+ } Mz2 ň ö » very small Nepleeting o% SO coso I ->tery small deglset) TE 12 m ( x ² + 29 30) + Į M² ta²g2 - T- [m mä²+zam żö & Mix² si ² + mo2a²

v V= -mga coso small oscillations. (1-coso) v= mga o 7 f 1-coro= 0% Page 5 writing Kit & P.E matrices, T į (m+ M) a² + masötma ox + mo²a2 T à (+ mo ma a ma 2 (m + M) os ma ma² maz 0 ma 2 v It -10 mga oz V 2 o v= mag O o 1 mag