When JEE Advanced Challenged Physics Lovers with "Torque by Time Varying Force" | PYQ 2018 скачать в хорошем качестве

When JEE Advanced Challenged Physics Lovers with "Torque by Time Varying Force" | PYQ 2018

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 a body of mass 1.0 kg at rest at the origin at time t = 0. A force F = (αt î + β ĵ) is applied on the body, where α = 1.0 N s⁻¹ and β = 1.0 N. The torque acting on the body about the origin at time t = 1.0 s is τ. Which of the following statements is (are) true? (a) |τ| = 1/3 N m (b) The torque τ is in the direction of the unit vector +k̂ (c) The velocity of the body at t = 1 s is 1/2( î + 2 ĵ) m s⁻¹ (d) The magnitude of displacement of the body at t = 1 s is 1/6 m. To solve this question, we start by understanding how the force acts on the body. Since the force varies with time, the acceleration of the body also changes with time. The motion is purely translational because the force acts directly on the particle. From the relation between acceleration and force, we can determine the velocity of the body by integrating the acceleration over time. Once the velocity as a function of time is known, we can find the position vector of the body by integrating again. Using the position and force vectors, we can then find the torque at a given instant, because torque is defined as the cross product of position and force. Let us now define two important physical quantities used in solving this problem. Definition 1: Torque Torque is the rotational effect of a force about a point or an axis. It depends on the magnitude of the force, the perpendicular distance between the line of action of the force and the point about which torque is measured, and the direction is given by the right-hand rule. In this question, torque determines how much rotational tendency the time-dependent force exerts about the origin. Definition 2: Displacement Displacement is the vector quantity that represents the change in position of a body. It gives both the shortest distance between the initial and final positions and the direction of motion. In this problem, displacement helps in determining the position vector of the particle when torque is evaluated. Now, let us discuss the theorems used in solving this question. Theorem 1: Newton’s Second Law in Vector Form According to Newton’s Second Law, the net force acting on a particle is equal to the rate of change of its linear momentum. For constant mass, this means the acceleration of the particle is directly proportional to the applied force. By integrating the acceleration over time, we obtain the velocity and displacement of the particle. In this question, this theorem is essential for finding the velocity of the body at any instant. Theorem 2: Relation Between Force, Torque, and Linear Motion The torque on a particle about a point is equal to the rate of change of its angular momentum about that point. The angular momentum, in turn, is defined as the cross product of the position and linear momentum vectors. This theorem connects translational motion with rotational effects, allowing us to calculate the torque from known values of force and position at a particular time. By combining these concepts, we can determine the correct statements. The direction of torque follows the right-hand rule and points along the positive z-axis. The time-dependent nature of the force causes a gradual increase in velocity and displacement, which we use to compute the torque magnitude. This problem elegantly combines the fundamentals of linear motion and rotational motion, showing how the two are interconnected. Understanding this concept is vital for mastering the chapter "System of Particles and Rotational Motion", especially for solving IIT-JEE Advanced questions that mix linear and angular quantities. #jeeadvanced #jeeadvance #iitjee