Posted by amena on Sunday, May 29, 2011 at 9:13am.
Let u be the static friction foefficient.
Linear Acceleration = a
= M[g*sin52/M -M*g cos52*u]/M
= g*[sin52 - u cos52]
Angular acceleration = alpha = a/R
= g [sin52 -ucos52)/R
(if there is no slipping)
= Ff*R/I < u*M*g*sin52*R/[(2/5)M R^2]
= (5/2)*u*g*sin52/R
Ff is the static friction force, which has a maximum value when slipping occurs.
To avoid slipping for a solid sphere, at that 52 degree slope angle,
(5/2)*u*g*sin52/R > g [sin52 -ucos52)/R
g's and R's cancel.
u*[(5/2)sin52 + cos52] > sin 52
u > 1/[(5/2) + cot 52] = 0.305
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