Question

A particle of mass ? slides along the inside of a smooth spherical bowl of radius ?. The particle remains in a vertical plane (the ??-plane). 

(a) Assuming that the bowl does not move: 

(i) Write down the Lagrangian, taking the angle ? with respect to the vertical direction as the generalized coordinate. 

(ii) Hence, derive the equation of motion for the particle. 

(b) Assuming that the bowl rests on a smooth horizontal table and has a mass ? and that the bowl can slide freely along the ?-direction: (i) Write down the Lagrangian in terms of the angle ? and the ?-coordinate of the bowl, ?. 

 (ii) Hence, derive the equations of motion for the system. 

(iii) Identify any cyclic coordinates in the system and interpret the corresponding conserved momenta. 

(iv) Using the expressions for the conjugate momenta, determine expressions for the generalized velocities in terms of the generalized coordinates and their conjugate momenta. 

Answer 1

CIASSMAte Dato: Page 0 D διαρό (a ③ Assuming that bowl is at rest, By definition, lagran gian is: T-V T= Img +8²8 2) O T T radius (as bowl is fixed) + m2 82 v= -mgx mga coso : 1=T-v L= 2 m 228 2 + marcoso

CIASSMAte Page Ü Equation of motion lagran gian Equation. Can be found using Euler- & ( 24. - d (8) As here o as generali Zed (o-ordinates d =0 20 at De 0 - mgrsino a mozó) = 0 - mgrsino je gosino +828 = më =0 2 mo o 0 g sino o b when bowl can slide freely, u > नी 00 particle 180-0 (720273-2003 COS (180-0) vector sum of two motion cas particle remains on spherical bowl)

CIASSMAte Date Pogo Now, Kinetic Energy KE Bout KE pastice 2 M 2 ² + I 글 m ((7202 +32-20 6 COS(180-6)) 03343/? As there is no motion ) is y direction, V =- mg o coso La T-V L Ź Mx² + 1 - 2 m (2202 +22 - 27 gæcose) +mgo cose tusing Euler- lagrange Equation: OL go d at () tmrox sino mgrsing -of ( mozý – mora cose) =0 mrosstho mgrsino mg2ě + mox coso se sinee = 0 0 je cose - gising - vp =0

CIASSMA te OLE di () 20 - 2 (M² Mai + må -mre coso ) ů (M+m) tm 82 Sino mrocoso =0 eu as lagrangian does not contain x tom, a x is cyclic Coordinate. mo coso = constant =k 12 : (Mtm) i OL = Px or conserved momenta. (in ů = px - mocoso (M+m)