^Insect crawling in a hemisphere
Consider an insect crawling up a rough hemisphere. As the insect crawl up the force pulling it down the
hemisphere i.e. mgsinθ increases where as normal
reaction N = mgcosθ & thus limiting friction on it decreases. Let at a height h, mgsinθ is just balanced by limiting friction.
After this height insect tends to slip back. Thus at maximum height reached by the insect.
mgsinθ = msmgsinθ or = tanθ,