Question

A narrow beam of length L = 2om and mass M=100 kg is connected to the ground by a hinge a distance 2L/5 from its end. The beam is free to rotate around the hinge without friction, and the beam forms an angle 8 = 50 with the horizontal. A clay ball with mass m=10 ky travels left with spoed v1 = 30m/s towards the top of the beam. A second clay ball with mass mg=3kg also travels left but with speedy 25 m/s towards the bottom of the beam 1 2L/5 19 a) What is the moment of inertia of the beam around the hinge? b) The clny balls collide and stick to the beam. The collisions occur simultaneously and the beat is initially stationary What is the final angular velocity of the beam, m, and y? Will the system rotate clockwise or counterclockwise around the linger

Answer 1

Length of beam, L=20m, mass, me lookg mi Mass pes unit length, l for m o la So. at so •6/ re dn.ge² . dna tida. distance from o moment of inestia, dI = Total moment of inestia - +345 I d'I 27 / 345 11 ge² dm. -22 X=-22 5 345 +345, 01, I < 11 alm d3 1245 on, I = 3 3 ge² da. -21 / ·[ (3) m [ m22 (-2)] 32 19 125 23. os, 1 IE 19 375

(a) 19 375 mi? I - b) Now , O='50, M, Ir 21 sino 31 sind. ly a sotations So, long 4= 30ml ma 3 kg , 25 ml pespendicular distance of m and from o lia Let, anticlockwise are positive Net initial angulas momentum, Li xe, Mauzy la mivix 31 MzYx 21 2 sind. [. 3m, o, 2 m2 Now, the balls stick to the sod, the of inestia Max (21) mix ( 21 J² mili sino 12 11 L sine 5 new moment I' It ㅏ 19 m = AM2 t gm, . 375 - 25 25 on, [/ H lôm 115 I'- + AM2+gn, gmy] + 22 25 final angulas relocity then using angulas momentum conses ration I'w= Li If , be OD, I'

Lsind [smile, - 2M24] 19 m + AM2 + 9m,] x 12 5x 15 25 on, W- 5 sind [ 3mie 2 M2 Uz - [ + 4M2toni] 19m 15 + on, 5xsin 50 x [ 3x10x30 - 2x 3 x 25 ) + gx10] 20 x [ 19x100 + 4x3 15 0, wa 628 sad/s As is positive to angulas relocity in anti-clockwise disection wa 0:628 sad's is