Adv Maths Question 37

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Question 37 A heavy particle lies on a smooth horizontal table and is attached to one end of a light inelastic string of length L. The other end of the string is attached to a point A on the circumference of the base of a vertical post which is fixed onto the table. The base of the post is a circle of radius a with its centre at a point O on the table. Initially, at time t = 0, the string is taut and perpendicular to the radius OA. The particle is then struck in such a way that the string starts winding round the post and remains taut. At a later time t, length a (< L) of the string is in contact with the post. Using cartesian axes, or otherwise, find the position and velocity vectors of the particle at time t in terms of a, L, and , and hence, or otherwise, show that the speed of the particle is (L a) . If the initial speed of the particle is v, show that the particle hits the post at time L 2/(2av). Discussion This is a familiar situation: it occurs often in Tom and Jerry cartoons; or you may perhaps have tried it yourself with a conker on a string. You have probably noticed that the conker rotates faster and faster as the string shortens. In fact, at the very last moment, the angular speed is infinite. Of course, this can only happen in an idealised situation, but it is not uncommon in such situations for things to become infinite in a finite time. For example, an ideal ball bouncing on the spot bounces an infinite number of times before coming to rest in a finite time. Do not be tempted to conserve angular momentum about the centre of the post. The line of the string does not pass through the centre of the post, so the tension in the string provides a couple about the centre which changes the angular momentum. The last paragraph involves integration.