A block of mass m is placed on a surface with a vertical...

A block of mass $m$ is placed on a surface with a vertical cross-section given by $y=\frac{x^{3}}{6} .$ If the co-efficient of friction is $0.5$, the maximum height above the ground at which the block can be placed without slipping is: (A) $\frac{1}{6} m$ (B) $\frac{2}{3} m$ (C) $\frac{1}{3} m$ (D) $\frac{1}{2} m$

A block of mass $m$ is placed on a surface with a vertical cross-section given by $y=\frac{x^{3}}{6} .$ If the co-efficient of friction is $0.5$, the maximum height above the ground at which the block can be placed without slipping is:

(A) $\frac{1}{6} m$
(B) $\frac{2}{3} m$
(C) $\frac{1}{3} m$
(D) $\frac{1}{2} m$

Key Concepts

Differentiation and Slope Analysis in Geometry

The slope of a curve at any point is given by its derivative, which is central to understanding how the geometry of the surface affects the forces acting on an object. In problems involving curved surfaces, the derivative not only gives the steepness of the incline but also informs the calculation of the normal and tangential components of gravitational force, thereby influencing conditions like maximum height or critical angles for static equilibrium.

Force Decomposition on Curved Surfaces

When dealing with non-linear or curved surfaces, force decomposition becomes more nuanced. The local geometry of the surface, characterized by its slope or derivative at a given point, plays a critical role in determining the orientation of the normal and tangential axes. This decomposition is essential for correctly applying Newton's laws of motion to assess the equilibrium state of an object on the surface.

Statics Equilibrium on Inclined Surfaces

This concept involves analyzing the conditions under which an object remains at rest on a surface. In the context of forces acting on an inclined or curved surface, it is essential to decompose the gravitational force into components parallel and perpendicular to the surface. The balance of these forces, along with the reaction forces from the surface, determines whether the object remains static or starts to slide.

Static Friction and its Maximum Value

Static friction is the force that prevents an object from sliding along a surface when a tangential force is applied. It has a maximum value, given by the product of the coefficient of static friction and the normal force. For an object to remain stationary, the component of the gravitational force along the surface must be less than or equal to this maximum frictional force, which sets a limit on the conditions under which the object can be safely placed without slipping.

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