Archive for January, 2011

Physics Friday 150

January 7, 2011

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?


Monday Math 149

January 3, 2011

What is the probability that two independently randomly chosen integers are mutually prime (have no common factor greater than 1)? The probability for four random integers? For n integers in general?