Question

A U-shaped vertical pipe of very small diameter is fixed on a trolley, as in figure. The pipe is made by a half circle ABC of radius R=20 [cm] and 2 rectilinear arms, one arm over the second one, at a height 2R up. The total mass of pipe and trolley is m = 1 [kg]. All frictions and viscosities are absent. A small sphere of same mass m is glued in the point C, in the upper arm. Initially the system has a horizontal velocity vo=20[m/s]. At the time t=0, the glue dries and the sphere detaches and falls down inside the pipe. Choose and show a reference system and find: glue B a) Which quantity (or quantities) is (are) conserved along the path CD, from the detaching to the exit of the sphere in D, explaining why is/are conserved b) The total kinetic energy Ek of the D system trolley+pipe+sphere when m is in C c) the velocity v, of the sphere when it is in D, on the lower arm, d) the velocity V, of the trolley when the sphere is in D, on the lower arm. А vo

Answer 1

'm' Page 1 mass glued at 'C' С. e R B R D Vo (a) Total energy is no or external mass glued at 'c? is conserved as there friction viscosi osity. Linear momentum of the s the system ats along x-direction is conserved due to the absence of force in the re-direction. (b) Ex= 1 m vo ² + 2m vo? Tipe + tronny Ek = mv2 = 1x 20²= 400 ] (C) Exit relocity of a mass m'at D= Velocity of the trolley when massim is VD lineal momentum (m+m) vo m VD VoVi at D g Conservation mão 2 Vo

Page 2 By energy conservation : ² (2m) ² + mg (2R) = 1 m (ge +_maya 2 1, 2+2 ZRg + Vo 2 2 2 (və+2Rg) = 2Vot voj + voz 400D + (480-200 - 4gR) r 2 2 VD +40 = Г. 2 = 0 ♡ Vo + 2 V. Vot (ava ZgR) Np - 2 vot / 40²-4X(065-296) 2 = - Vot 0.5 =) Vo 1.414&gr V 8 - Votto (d) We have, V 2 Vo = VD Yo = 2Vot to Vot 1.414 1.414 JgR VD