Ball on string, conservation
I am reviewing physics by going through "fundamentals of physics." I am working on a problem where a massless string of length L is connected to a ball, and is released from horizontal. directly below the top end of the string a distance d below is a peg. The distance between the peg and the bottom of the ball's motion is r (or Ld). The problem is: "show that, if the ball is to swing completely around the fixed peg, then d>3L/5."
My approach is the following: the kinetic energy (K1) at the bottom of its motion is mgL (by conservation). For the ball to continue moving at the top (a distance r above the peg) it's kinetic energy must be greater than 0. By conservation, K1=K2+U, where U is the potential energy above the peg. Setting U=mg(2r), and K2=K1U>0, then K1>U, or mgL>mg(2r). Since r=Ld, mgL>2mgL2mgd. Then, 2mgd>mgL, and finally: d>L/2. This is not the answer d>3L/5. Help?
