E) vavc
18. an insect crawls up a hemispherical surface
very slowly. the coefficient of friction
between the insect and the surface is 5. if the
line joining the center of the hemispherical
surface to the insect makes an angle a with
the vertical, the maximum possible value of a
is given by
(a) cota = 3
(b) tana = 3
6) seca = 3
(d) coseca = 3

b) tan a = 3

Explanation:

Draw a free body diagram (see attached).

There are three forces acting on the insect.  Weight downwards, normal force towards the center of the hemisphere, and friction tangent to the surface.

Sum of the forces in the radial direction:

∑F = ma

N − mg cos α = 0

N = mg cos α

Sum of the forces in the tangential direction:

∑F = ma

μN − mg sin α = 0

Substituting:

μ(mg cos α) − mg sin α = 0

μ cos α − sin α = 0

μ cos α = sin α

μ = tan α

The maximum possible value of the angle is such that its tangent is equal to the coefficient of friction.

that tool is called an altimeter which measures your altitude which can be used to tell which atmosphere you are currently in, including the stratosphere.

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