This Is Why I Love JEE Advanced! It Can Surprize Anyone With An Easy But Unexpected Question

In this Physics video in Hindi for the chapter "System of Particles and Rotational Motion" of Class 11, we discussed a Previous Years’ Question of IIT-JEE Advanced.

The question states:
Consider regular polygons with number of sides n = 3, 4, 5, ... as shown in the figure. The centre of mass of all the polygons is at height h from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus of the center of mass for each polygon is Δ. Then Δ depends on n and h as
(a) Δ = h sin²(π/n),
(b) Δ = h tan²(π/2n),
(c) Δ = h(1/cos(π/n) − 1),
(d) Δ = h sin(2π/n).
[IIT-JEE Advanced 2017]

This question beautifully combines rotational motion, geometry, and the motion of the center of mass. When a polygon rolls on a surface about one of its vertices, the center of mass rises and falls periodically as each vertex becomes the new point of contact. The problem asks us to find the maximum rise in height (Δ) of the center of mass as the polygon transitions from one vertex to the next.

To solve this question, we analyze the geometry of the polygon and how the center of mass moves relative to the point of contact. When a regular polygon rolls about a vertex without slipping, the center of mass traces an arc around that vertex. The height difference between the lowest and highest positions of the center of mass depends on the angle subtended at the center of the polygon and the height h. By relating the geometry of the situation with the number of sides n, we find that Δ depends on both h and n in a trigonometric manner.

One of the key physical quantities involved in this question is the Centre of Mass. The centre of mass of a body is the point where the total mass of the system can be considered to be concentrated for the purpose of analyzing translational motion. In the case of the rolling polygon, the motion of this point determines how the body as a whole rises and falls during rotation.

Another important concept here is Rolling Without Slipping. Rolling without slipping means that the body rotates in such a way that the point of contact with the surface has zero relative velocity with respect to the surface. This condition ensures that there is a pure rolling motion, allowing the geometric relationships between rotation and translation to hold accurately.

This question is an excellent example from the chapter "System of Particles and Rotational Motion" because it integrates rotational dynamics, energy concepts, and geometric reasoning. It challenges students to visualize the motion of the center of mass during rolling and to express its height change using trigonometric relationships. Such problems are crucial for mastering IIT-JEE Advanced level conceptual understanding.

Through this video, we explored the geometrical logic, physical principles, and systematic approach required to solve this unique IIT-JEE Advanced question. Students will gain a deep conceptual grasp of rotational motion and geometric transformations involved in rolling motion — topics central to System of Particles and Rotational Motion and essential for IIT-JEE Advanced success.

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