Problem 5. The particle P with mass mp is supposed to travel along a circular path...

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Problem 5. The particle P with mass mp is supposed to travel along a circular path from a point A to a point B in a vertical plane (i.e. take a change of gravitational potential energy into account). a) In the first construction depicted in Fig. 4 (left), a rigid rod with length R and negligible mass is piv- oted at point O and carries the particle P. The particle P will travel along a circular path about point O when it is released from point A with the initial velocity VA. Assume that friction can be neglected, and employ the conservation of energy to determine the minimum initial velocity VA required such that the particle will reach point B. b) In the second construction depicted in Fig. 4 (right), the rod got removed, and the circular path of the particle P shall be guaranteed by the circular construction with radius R about point O. To keep the particle P on the circular path, the axial force N between the particle and the circular construction must not turn negative. Assume that friction can be neglected, and employ the conservation of energy to determine the minimum initial velocity V required such that the particle will reach point B. c) Assume that mp = 5 kg and R = 2m. Give numerical results for VA obtained from Parts a) and b), compare the velocities, and explain briefly if the result is physically reasonable. B R O 60 A B R A Figure 4: Two constructions to keep a particle P on a circular path. Problem 5. The particle P with mass mp is supposed to travel along a circular path from a point A to a point B in a vertical plane (i.e. take a change of gravitational potential energy into account). a) In the first construction depicted in Fig. 4 (left), a rigid rod with length R and negligible mass is piv- oted at point O and carries the particle P. The particle P will travel along a circular path about point O when it is released from point A with the initial velocity VA. Assume that friction can be neglected, and employ the conservation of energy to determine the minimum initial velocity VA required such that the particle will reach point B. b) In the second construction depicted in Fig. 4 (right), the rod got removed, and the circular path of the particle P shall be guaranteed by the circular construction with radius R about point O. To keep the particle P on the circular path, the axial force N between the particle and the circular construction must not turn negative. Assume that friction can be neglected, and employ the conservation of energy to determine the minimum initial velocity V required such that the particle will reach point B. c) Assume that mp = 5 kg and R = 2m. Give numerical results for VA obtained from Parts a) and b), compare the velocities, and explain briefly if the result is physically reasonable. B R O 60 A B R A Figure 4: Two constructions to keep a particle P on a circular path.