JEE Advanced Constructed This "Leaning Ladder" Problem to Test "Common Sense" With Knowledge скачать в хорошем качестве

JEE Advanced Constructed This "Leaning Ladder" Problem to Test "Common Sense" With Knowledge

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 A ladder of mass m is shown leaning against a wall, making an angle θ with the horizontal floor. The coefficient of friction between the wall and the ladder is μ₁, and that between the floor and the ladder is μ₂. The normal reaction of the wall on the ladder is N₁ and that of the floor is N₂. The ladder is just about to slip, and we have to determine the correct relationship between these quantities. The given options include various cases depending on which frictional forces are active at the wall and the floor. To solve this question, we start by analyzing all the forces acting on the ladder. The weight of the ladder acts vertically downward through its center of gravity. The wall exerts both a horizontal normal reaction and a vertical frictional force on the ladder, while the floor exerts a vertical normal reaction and a horizontal frictional force. Since the ladder is in static equilibrium, both the net horizontal and vertical forces must be zero, and the net torque about any point must also be zero. These conditions help us derive the required relation among the forces when the ladder is on the verge of slipping. Now, let’s define two important physical quantities that are essential to solving this problem. Definition 1: Frictional Force Frictional force is the resistive force that acts between two surfaces in contact and opposes their relative motion or the tendency of such motion. In this problem, there are two frictional forces acting — one at the wall and one at the floor. The frictional force at the wall prevents the top of the ladder from sliding down, while the frictional force at the floor prevents the bottom of the ladder from sliding away from the wall. Definition 2: Normal Reaction Normal reaction is the contact force exerted by a surface perpendicular to the object in contact. It prevents the object from penetrating into the surface. In this ladder problem, the wall exerts a normal reaction horizontally on the ladder, while the floor exerts a normal reaction vertically upward. These normal reactions play a crucial role in determining the frictional forces, since the limiting friction at each surface is proportional to the respective normal reaction. Let us now recall two key theorems used in solving this question. Theorem 1: Conditions for Static Equilibrium A rigid body is said to be in static equilibrium when the vector sum of all the external forces acting on it is zero, and the algebraic sum of all torques acting on it about any axis is also zero. In this ladder problem, the equilibrium of forces ensures that the ladder neither slides nor moves vertically, and the torque condition ensures that it does not rotate about its center or any other point. Theorem 2: Limiting Friction Theorem When a body is on the verge of slipping, the frictional force attains its maximum possible value, called the limiting friction. The limiting friction is directly proportional to the normal reaction, with the constant of proportionality being the coefficient of friction. In this problem, we apply the condition of limiting friction at both the wall and the floor to determine the relationship between μ₁, μ₂, N₁, and N₂ when the ladder is about to slip. By applying these theorems together, we analyze the balance of forces and moments. The horizontal equilibrium connects the frictional force at the floor with the normal reaction at the wall, while the vertical equilibrium connects the frictional force at the wall with the normal reaction at the floor. Using the torque condition about the point where the ladder touches the floor, we eliminate one variable and find the correct relationship between the forces. The final result depends on whether friction acts at both the wall and the floor or only at one of them. This question is a perfect demonstration of the interplay between linear and rotational equilibrium in the chapter "System of Particles and Rotational Motion". It also highlights how multiple frictional forces act simultaneously to maintain static equilibrium. Such questions are common in IIT-JEE Advanced examinations and test the understanding of both force and torque balance. The concept of limiting friction and equilibrium of forces is fundamental to solving many ladder-type problems in the "System of Particles and Rotational Motion" chapter. #jeeadvanced #jeeadvance #iitjee