Question

if the coefficient of friction between an insect and bowl surface is u and the radius of the bowl is r ,the maximum height to which the insect can crawl in the bowl

Answer 1

From the attached figure, let say the insect can crawl upto height h, where the tangential component of weight is equal to the limiting friction.

Hence,

umgcosФ = mgsinФ

=> tanФ = u

=> cos Ф = $\frac{1}{\sqrt{u^2+1} }$

from the figure we can see that,

cosФ = $\frac{r-h}{r}$

=> h = cosФ(1-r)

=> h = $\frac{1-r}{\sqrt{u^2+1} }$

Answer 2

Given :-

Coefficient of Friction = μ

R = mg CosØ

F = mg SinØ

tanØ = F/R

tanØ = μ

μ = √(r²-y²)/y

μ²y² = r² - y²

y²(μ² + 1) = r²

y = √r²/(μ²+1)

y = r/√(μ²+1)

Again,

h = r - y

h = r - r/√(μ²+1)

h = r[1-1/(μ²+1)]

Hence,

The height crawled by insect = h = r[1-1/(μ²+1)].