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Aippo Mar 23rd 2016 08:40 PM

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 L-d). 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=K1-U>0, then K1>U, or mgL>mg(2r). Since r=L-d, mgL>2mgL-2mgd. Then, 2mgd>mgL, and finally: d>L/2. This is not the answer d>3L/5. Help?

ChipB Mar 24th 2016 08:12 AM

The piece you are missing is that the ball's velocity as it swings around the peg must actually be significantly greater than 0. At the top of its arc its centripetal acceleration must be at least as great as g - otherwise the ball would fall and not continue around in an arc. Stated another way - there must always be a positive tension in the string.

Aippo Mar 25th 2016 05:23 PM

Many thanks
 
Thank you so much. This was racking my brain and now it makes perfect sense.


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