Question

A mass m is attached firmly to the end of a massless stick of length 6. The other end of the stick is fixed to the wall at x = 0 by a hinge and pivots up and down frictionlessly. The hinge is a height h above the floor. Two vertical massless springs (both with equilibrium length h) of force constants ki and k2 support the stick at points 1 and 22, respectively. (a) Write the Lagrangian and the Euler-Lagrange equation for the mass. (b) What is the equilibrium position of the mass? (c) Make a small angle approximation and calculate the frequency of oscillation? h ------ x X2

Write the Lagrangian and Euler-Lagrangian equation for the mass.

What is the equilibrium position of the mass?

Make a small angle approximation and calculate the frequency of oscillation.

Answer 1

* Solution :- S06 j(x, y) bsin a -- P.Ei=0 (a) Co-ordinates of mass'm', x = bcos o i = -b sino é y = h+ b sino = ý = b cosos The compression / h elongation of springs are Ys and yz. > o x x yo } xitano = y1 A = (x, th + xq tano) B = ( 29, h + x2 tano) * X K2 coso se }} * =1(22 cosa je – det k ₂ - =% tano .: Kinetic energy of the system, kiE.(1), Tamät +j2) = 1m [Eb sino ' + (bcoso o)?] = m ( 6? sino + 6? cos?o) 52 T = 1 / 2 m b² o Potential energy, v= 2 kg 4 + 2 Kz y + mgb sino V= kg (24 tano)* + [ kg (32 tanio)? + mg bsino *V = I (K1 ** + Kx x*) tano + mgb sino > lagrangian, L= T-V

L = √ mb²8² - / (K1 X Y + K2 KE) tanko - mgbsine (s, and se Jare constants) Lagrangian equation of motion, a caus) olet ( mb?ó) = -4 (K4X] + Ks 27). e tamosecüo - mgb cos o = (kg 84? + K3 X32). 4-sine teme - mgo coso mot ö = - (K1 x} + K2 839). sino - mgb coso → mb2 ö = -(K4 x} + K2 *33) sino - mgb coso 3 ano. sino 1 os a coso cosso Cos 30 (b) equillibrium position of mass : Os : at 0=0, e o loco o "ok" 4060 O → + -(KL.2? + K2 222 ) sind - mgb coso, = 0 cos3o (K, 3ị + K, ) An, = - mºb 18+. for small angles, sino ao coso = 1-02 → cosło = 1 → (Ke x2 + Ky x2).= -mgb = -mgb Kq wf + kg uz

(C) for small angles, sino co, cos o -1 - mb²ö = - (k2 x ² + Kg x ²) o-mgb + 6 = - ( 3 + %) - m 62 → = -(Ke 2? +Kz33) co + mgb m 62 K1941€ + kz 293 define, o=0+ mgb K x + Kx X2 + P = -(Ke 72 +K222) o' - mb² comparing it with SHM equation, -W232 → w²= kg af + K₂ xq lw=Kq x² + K₂ 2₂² тв I N mb²