JEE Advanced Tested 9 Concepts of Rotational Motion in "Falling Rod" Problem in 2019 Paper скачать в хорошем качестве

JEE Advanced Tested 9 Concepts of Rotational Motion in "Falling Rod" Problem in 2019 Paper

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 from 2019 paper. The question states:- A thin and uniform rod of mass M and length L is held vertically on a floor with large friction. It is released from rest so that it falls by rotating about its contact point with the floor without slipping. Which of the following statement(s) is/are correct, when the rod makes an angle 60° with vertical? [g is the acceleration due to gravity] (a) The angular acceleration of the rod will be 2g/L (b) The normal reaction force from the floor on the rod will be Mg/16 (c) The radial acceleration of the rod’s center of mass will be 3g/4 (d) The angular speed of the rod will be √(3g/2L) The given options involve calculating the angular acceleration of the rod, the normal reaction force from the floor, the radial acceleration of the center of mass, and the angular speed of the rod at that instant. To solve this question, we begin by analyzing the motion of the rod as it starts falling. Since the rod rotates about its lower end without slipping, the point of contact acts as an instantaneous axis of rotation. The gravitational force acts at the center of mass, located at the midpoint of the rod, and produces a torque about the contact point. This torque causes an angular acceleration, which can be found using the rotational analog of Newton’s second law. By applying the energy conservation principle, we can also determine the angular speed when the rod has rotated through a given angle. Now let us define some important physical quantities that are used in solving this question. Definition 1: Angular Acceleration Angular acceleration is the rate of change of angular velocity with respect to time. It describes how quickly the rotational speed of a body changes when a torque acts upon it. In this case, the torque due to the weight of the rod about the contact point produces angular acceleration, which determines how fast the rod’s rotational motion increases as it falls. Definition 2: Normal Reaction Force Normal reaction is the perpendicular contact force exerted by a surface on a body resting or rotating upon it. In this problem, the floor exerts an upward normal force at the contact point, balancing part of the gravitational force and providing the necessary centripetal force for the circular motion of the rod’s center of mass. Let us now discuss the theorems that are fundamental in solving this question. Theorem 1: Rotational Form of Newton’s Second Law The rotational form of Newton’s second law states that the torque acting on a rigid body about an axis is equal to the rate of change of its angular momentum about that axis. This is mathematically represented as torque equals the moment of inertia times angular acceleration. In this question, we apply this theorem by taking the torque due to gravity about the contact point and relating it to the rod’s angular acceleration using its moment of inertia about that point. Theorem 2: Conservation of Mechanical Energy The principle of conservation of mechanical energy states that in the absence of non-conservative forces, the total mechanical energy of a system remains constant. In this problem, since there is no slipping and friction does not do work, the loss in potential energy of the rod’s center of mass as it falls is completely converted into rotational kinetic energy. This allows us to calculate the angular speed of the rod when it reaches the specified angle. To find the radial acceleration of the center of mass, we note that as the rod rotates, the center of mass follows a circular path about the contact point. The radial acceleration is directed toward the point of contact and depends on the square of the angular speed and the distance of the center of mass from the axis of rotation. The normal reaction from the floor can then be found by balancing the vertical and horizontal forces acting on the rod at that instant. This question beautifully combines multiple core concepts of the chapter "System of Particles and Rotational Motion" — torque, rotational inertia, angular acceleration, and conservation of energy. It is a typical example of how IIT-JEE Advanced tests conceptual clarity rather than mere memorization. By carefully analyzing rotational dynamics and the conditions for rolling without slipping, we can solve such problems with confidence. Understanding this question strengthens one’s grasp over the chapter "System of Particles and Rotational Motion" and prepares students well for similar complex problems in the IIT-JEE Advanced examination. #jeeadvanced #jeeadvance #iitjee