Physics Friday 150

Consider a skater of mass m on a smooth skating rink (so that we may ignore friction in this problem) with a cylindrical pillar, of radius R on the rink. A rope (of negligible mass compared to the skater) has one end fastened to the column, and extends straight out tangentially from the column for a length L. If the skater grabs the end of the rope while having a velocity v perpendicular to the rope, and then spirals inward, the rope winding onto the column. If the rope remains straight and taut throughout the spiral, what will the skater’s speed be upon reaching the pillar? In what direction?

Treating the rope as massless, we see that the part of the rope wound onto the column doesn’t move, and the point where it winds on has motion with no component along the length. Thus, we see the column does no work on the rope, and so the rope can do no work on the skater. Thus, the kinetic energy of the skater is unchanged, and so is his speed; he collides with the column with a speed v. Since the rope does no work on the skater, we see the force it exerts on the skater must be perpendicular to the skater’s path at all times (so that F·ds=0), and since the rope exerts force only parallel to its length, we see the skater’s velocity (tangent to his path) must always be perpendicular to the rope. Thus, when he reaches the column, the rope is fully wound, and so tangent to the column. Therefore, the skater’s velocity is normal to the column at impact; he hits it straight on. (Specifically, the skater’s path is the involute of the circle).


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