Q. A particle is placed at rest inside a hollow hemisphere of radius R . The coefficient of friction between the particle and the hemisphere is 1/✓3 . The maximum height upto which the particle can remain stationary is
(1)R/2
(2)(1- ✓3/2) R
(3)(✓3/2) R
(4)3R/8

Sir/mam please explain the answer of this question and then after please solve it numerically .

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Solution

on balancing forces we have

normal rxn. N = mg.cosx

f = mg.sinx

where f = friction coeff. * N

f = mgcosx /3^1/2
mgsinx = mgcosx /3^1/2
so tan x = 1/3^{1/2

x = 30 degrees
height = R - Rcos30}

so height at which particle remains stationary is ( 1 - 3^1/2 /2 )R

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on balancing forces we have normal rxn. N = mg.cosx f = mg.sinx where f = friction coeff. * N f = mgcosx /31/2 mgsinx = mgcosx /31/2 so tan x = 1/31/2 x = 30 degrees height = R - Rcos30 so height at which particle remains stationary is ( 1 - 31/2 /2 )R

on balancing forces we have

normal rxn. N = mg.cosx

f = mg.sinx

where f = friction coeff. * N

f = mgcosx /3^1/2
mgsinx = mgcosx /3^1/2
so tan x = 1/3^{1/2

x = 30 degrees
height = R - Rcos30}

so height at which particle remains stationary is ( 1 - 3^1/2 /2 )R