22nd_IPhO_1991 - THEORETICAL PROBLEMS Problem 1 The figure...

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T H E O R E T I C A L P R O B L E M S Problem 1 The figure 1.1 shows a solid, homogeneous ball radius R . Before falling to the floor its center of mass is at rest, but the ball is spinning with angular velocity ω 0 about a horizontal axis through its center. The lowest point of the ball is at a height h above the floor. When released, the ball falls under gravity, and rebounds to a new height such that its lowest point is now ah above the floor. The deformation of the ball and the floor on impact may be considered negligible. Ignore the presence of the air. The impact time, although, is finite. The mass of the ball is m , the acceleration due the gravity is g , the dynamic coefficient of friction between the ball and the floor is μ k , and the moment of inertia of the ball about the given axis is: I = 5 2 2 mR You are required to consider two situations, in the first, the ball slips during the entire impact time, and in the second the slipping stops before the end of the impact time. Situation I: slipping throughout the impact. Find: a) tan θ , where θ is the rebound angle indicated in the diagram; b) the horizontal distance traveled in flight between the first and second impacts; c) the minimum value of ω 0 for this situations. Situation II: slipping for part of the impacts. Find, again: a) tan θ ; b) the horizontal distance traveled in flight between the first and second impacts. Taking both of the above situations into account, sketch the variation of tan θ with ω 0. Problem 2 In a square loop with a side length L, a large number of balls of negligible radius and each with a charge q are moving at a speed u with a constant separation a between them, as seen from a frame of reference that is fixed with respect to the loop. The balls are arranged on the loop like the beads on a necklace, L being much greater than a , as indicated in the figure 2.1. The no conducting wire forming the loop has a homogeneous charge density per unit length in the in the frame of the loop. Its total charge is equal and opposite to the total charge of the balls in that frame.
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Consider the situation in which the loop moves with velocity v parallel to its side AB (fig. 2.1) through a homogeneous electric field of strength E which is perpendicular to the loop velocity and makes an angle θ with the plane of the loop. Taking into account relativistic effects, calculate the following magnitudes in the frame of reference of an observer who sees the loop moving with velocity v : a) The spacing between the balls on each of the side of the loop, a AB , a BC , a CD , y a DA . b) The value of the net charge of the loop plus balls on each of the side of the loop: Q AB , Q BC , Q CD y, Q DA c) The modulus M of the electrically produced torque tending to rotate the system of the loop and the balls. d)
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22nd_IPhO_1991 - THEORETICAL PROBLEMS Problem 1 The figure...

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