Question

Question 2 A pendulum is formed by suspending a mass m from the ceiling, using a spring of unstretched length lo and spring constant k (Figure 2). The pendulum moves in a fixed verticał plane. Choose x and y as the generalised coordinates. Set up the Lagrangian. Write down the equations of motion for the generalised variables. [4 marks] X Figure 2

Answer 1

Qe , Given Information is to and Spring l to thie system adle M Н. A pendulum consist of mass in cuuquet Length Constantak generalised Co-ordinate for LEO Spring makes with the vertical direction binetic energy of the system ²m m (i+122) energy CV)s-mg coso +. The Lagrangion of the system T = potential 3.50 +1 ak (-1) Le T-v к С.-44)- no Lam Citlich umge coso - 8%, there your dissipative force acting nergy of the system is conserved. the temitonion of the system T+V

H = h 2 m (inteo) - mgl coso- tk (d-do) ² Any Genenalized co-ordinate , pa dd dq 9 pe= IL IL and po dh do am t. am 12 Now, cruation , d dq Il 39 Ha de mi lehet team J-mgd caso tof KCd-de) PZ .) - mgd coso + 1 kk (d-10) Lagrange's ( hu ) putting L&o in place of "å Sant Cugat) Solving get më me or - mg me i famio +mgl thua, 3 at .) - dd do we coso tk (dodo) ao sino=0

t tlemiton een for career of motion ) foot generalized co-ordinate ?', for '<' -ě » -PL IH IZ IH IL ЭН IP 2 ЭН Ipd foro du do - po and dH d po - from these can's it is obtained PL (po ) - mg coso tk Clado) тя i= PL m po -mgad sino Po те- Thank you.