A body starts rolling down an inclined plane , the top half is rough . Find the ratio of the force of friction and weight of the body if the body is brought to rest just when it reaches the bottom , the angle of the plane being `30^(@)` .

Explanation

AB is an inclined plane of length 2 l with center at C
The top half AC of the plane is smooth and the lower half CB is rough Let F be force of friction in this part
for the top half of inclined plane
` u = 0 , a = g sin 30^(@) = (g)/(2) , s = l , upsilon = ` ?
As ` upsilon^(2) - u^(2) = 2 a s `
` :. upsilon^(2) - 0 = 2 ((g)/(2))l or upsilon= sqrt(g l `
For the bottom half of inclined plane ,
` u = sqrt(g l) , upsilon = 0 , s = l , a =` ?
As ` upsilon^(2) - u^(2) = 2 a s `
` :. 0 - g l = 2 a l , a = - (g)/(2) `
In this part , acc . ` = ("net force")/("mass")`
`a = (mg sin 30^(@) - F )/(m)`
` -(g)/(2) = g xx (1)/(2) - (F)/(m) or (F)/(m) = (g)/(2) + (g)/(2) = g`
or `F = mg :. (F)/(mg) = 1` .