June 13th, 2017, 06:03 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 258 Thanks: 23 Math Focus: Number theory  Twosphere pendulum
A ball of diameter D, inside a spherical shell of radius D, both with adequate friction at initial contact between equators, act as a pendulum with what period?
Last edited by Loren; June 13th, 2017 at 06:08 PM. 
June 14th, 2017, 03:38 AM  #2 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,049 Thanks: 680 Math Focus: Physics, mathematical modelling, numerical and computational solutions 
Um... same period as a regular pendulum with a mass equal to the total mass of the ball and shell? Not sure...

June 14th, 2017, 05:29 AM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,941 Thanks: 2267 Math Focus: Mainly analysis and algebra 
A diagram that illustrates how it acts as a pendulum might be useful.

June 14th, 2017, 06:40 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 17,919 Thanks: 1386  I think the shell is static, making its mass irrelevant. The question indicates that the ball rolls rather than slips, but without losing energy. Google finds several worked examples of this type of "pendulum", but they don't allow for the large angle of oscillation specified in this problem.

June 14th, 2017, 07:35 AM  #5  
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,049 Thanks: 680 Math Focus: Physics, mathematical modelling, numerical and computational solutions  Quote:
 
June 14th, 2017, 08:15 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 17,919 Thanks: 1386 
Assuming that the shell is hemispherical, one can reasonably assume that the ball starts as illustrated below. Pendulum.PNG 
June 14th, 2017, 03:02 PM  #7 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 258 Thanks: 23 Math Focus: Number theory 
Would a twodimensional, frictionless pendulum simplify the OP (now no slipping)? The ball would initially be released where the equators meet. Thanks for the diagram, skipjack. 
June 15th, 2017, 12:24 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 17,919 Thanks: 1386 
Do you mean that the ball doesn't "roll" at all?

June 15th, 2017, 08:30 AM  #9 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 258 Thanks: 23 Math Focus: Number theory 
I submit two "extremes"; the ball sliding without friction, or the ball rolling with just enough friction to cause it not to slide.


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