An insect crawls up a hemispherical surface very slowly. The...

An insect crawls up a hemispherical surface very slowly. The coefficient of friction between the surface and the insect is $1 / 3$. If the line joining the centre of the hemispherical surface to the insect makes an angle $\alpha$ with the vertical, the maximum possible value of $\alpha$ is given by (Figure can't copy) (a) $\cot \alpha=3$ (b) $\tan \alpha=3$ (c) $\sec \alpha=3$ (d) $\operatorname{cosec} \alpha=3$

An insect crawls up a hemispherical surface very slowly. The coefficient of friction between the surface and the insect is $1 / 3$. If the line joining the centre of the hemispherical surface to the insect makes an angle $\alpha$ with the vertical, the maximum possible value of $\alpha$ is given by
(Figure can't copy)
(a) $\cot \alpha=3$
(b) $\tan \alpha=3$
(c) $\sec \alpha=3$
(d) $\operatorname{cosec} \alpha=3$

Key Concepts

Static Equilibrium

Static equilibrium is the condition in which a body remains at rest because all the forces and moments acting on it balance out. This concept is fundamental when analyzing situations where forces need to cancel each other, ensuring there is no net acceleration.

Friction

Friction is the resistive force that acts between two surfaces in contact, opposing relative motion. Specifically, the maximum static frictional force is given by the product of the coefficient of friction and the normal force, setting a threshold for the onset of slipping.

Force Resolution

Force resolution is the technique of breaking down forces into components along convenient directions, such as parallel and perpendicular to a surface. This method simplifies the analysis of forces acting on bodies, especially when dealing with inclined or curved surfaces.

Normal Force

The normal force is the reactive force provided by a surface that acts perpendicular to the contact plane. It plays a critical role in frictional problems, as the maximum frictional force depends directly on the magnitude of the normal force.

Transcript

00:01 Hello everyone we are going to understand this question here in the question it is given that an insect is moving in on the spherical surface and coefficient of friction is given 1 by 3 given coefficient of friction of spherical surface coefficient of friction of spherical surface mu is equal to 1 by 3 and angle which it is making angle made by the insect alpha is equal to so the angle is made by the insect is alpha when it reached at this point then we have to find the maximum possible value of alpha so let's draw the fvd this angle is alpha when we will draw the tangent on at this point then this angle is 90 degree now with the horizontal, when we will draw the tangent, then and we will make this.

01:56 And then we can see this is the type of incline plane.

02:02 This angle is 90 minus alpha, so this angle will be alpha.

02:08 Again we can draw a fresh diagram.

02:13 This angle is alpha.

02:14 Insect is at this position.

02:16 So, weight of the insect is vertically downward.

02:22 M into g if we will take the component then component perpendicular to the surface is mg cos theta sorry mg cos alpha mg phos alpha and along to the inclined plane that is mg sine alpha and friction force will be in upward direction that is f s now applying the force balance equation from the fvd.

02:58 Now applying force balance equation from above fvd.

03:23 So we can write friction force fs is equal to muo into n.

03:33 Here value of mu is 1 upon 3 and value of n is mg cos alpha mg cos alpha...

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