A marble of mass m = 734 g and radius r = 18.4 cm rolls along the looped rough t

Question

A marble of mass m = 734 g and radius r = 18.4 cm rolls along the looped rough track of figure below. The loop has a radius of R = 1.98 m. What is the minimum value of h if the marble is to reach the highest point of the loop without leaving the track?

A marble of mass m = 734 g and radius r = 18.4 cm rolls along the looped rough track of figure below. The loop has a radius of R = 1.98 m. What is the minimum value of h if the marble is to reach the highest point of the loop without leaving the track?

Explanation / Answer

initial energy = final energy
m.g.h = m.g.h' + 1/2 m.v^2 + 1/2 I.w^2

h' = 2R = 2 * 1.98 =3.96 m
I for marble = 2/5.m r^2 = 9.94 * 10^-3
w = v/r

m.g = m.v^2 /R
v^2 = g.R

v = 4.407 m /sec

so w = 4.407 / 0.184 = 23.95

Now

m.g.h = m.g.h' + 1/2 m.v^2 + 1/2 I.w^2

susbtitute all the above values

h = 5.346 m

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