An insect crawls up a hemispherical surface very slowly (see fig.). The coefficient of friction between the insect and the surface is 1/3. If the line joining the centre of the hemispherical surface to the insect makes an angle $α$ with the vertical, the maximum possible value of $α$ is given by [2001]

Question

Smart Achievers · Accepted Answer

(1)

The two forces acting on the insect are $m g$ and $N$ .

Two components of $m g$ are

$m g \cos α balances N$

$m g \sin α is balanced by the frictional force.$

$∴$ $N = m g \cos α$

$f = m g \sin α$

But, $f = μ N = μ m g \cos α$

$∴$ $μ m g \cos α = m g \sin α \Rightarrow \cot α = \frac{1}{μ} \Rightarrow \cot α = 3$

Solution

1 $\cot α = 3$

2 $\tan α = 3$

3 $\sec α = 3$

4 $cosec α = 3$

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Solution

1 cotα=3

2 tanα=3

3 secα=3

4 cosecα=3

Ans. (1) The two forces acting on the insect are mg and N. Two components of mg are mgcosα balances N mgsinα is balanced by the frictional force. ∴ N=mgcosα f=mgsinα But, f=μN=μmgcosα ∴ μmgcosα=mgsinα⇒cotα=1μ⇒cotα=3

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1 $\cot α = 3$

2 $\tan α = 3$

3 $\sec α = 3$

4 $cosec α = 3$