# Q & A: Minimum velocity for a loop-the-loop pendulum

Q:
What is the minimum velocity to be given to the bob of a pendulum of length 'l' which is suspended normally so that it makes a complete vertical rotation? Is it rootover 4gl or rootover 5gl? If it is (root)4gl then what is (root)5gl for and if it is (root)5gl,then what is (root)4gl for?In some book I saw (root)4gl and in some, (root)5gl..
- Shawn (age 16)
India
A:

Hi shawn,

Both answers are right!   It depends on what sort of pendulum you are dealing with.   If the pendulum is made with a weight at the end of a mass-less rod then you get v2 = 4gL  This is easily obtained by equating the kinetic energy at the bottom, 1/2 mv2, to the potential energy at the top, 2mgL.     However, if you suspend the mass on a flexible string then the string cannot support any compressive force and will crumple before it gets to the top due to the gravitational force.  If you want to prevent this you need some extra velocity in order to have a centrifugal  force  counteracting that of gravity at the top.   This is mv2/L = mg.   Adding this needed extra energy will give you the other answer v2 = 5gL.

LeeH

(published on 10/05/2014)