Rolling Motion and Torque Analysis of a Spool: Solving Advanced Physics Problem

Welcome to this in-depth tutorial on rotational mechanics! In this video, we solve and theoretically analyze an advanced physics problem involving a spool rolling without slipping on a horizontal surface. 📊 Problem Statement: A spool has a mass of 2\text{ kg}, an inner radius R_1 = 3, and an outer radius R_2 = 5The radius of gyration about the axis of the spool is K = 4 cm A constant horizontal force of 5 N is applied to the free end of a massless thread wrapped around the inner cylinder of the spool (as shown in figure-5.150). If the spool rolls without slipping, calculate the linear acceleration along the horizontal surface, and determine the minimum coefficient of static friction required to prevent slipping. 🔍 Theoretical Analysis & Concepts: Free Body Diagram (FBD) & Force Breakdown: Applied Force (F): A horizontal pulling force of 5N acts at the inner radius R_1, directed horizontally to the right. Normal Reaction (N) and Weight (Mg): Vertical forces balance out such that N = Mg, where M = 2kg and g approx 9.8m/s^2. Friction Force (f): Acts at the point of contact between the outer radius (R_2) and the ground. Depending on the torque direction, the friction force ensures pure rolling and helps govern the net translational acceleration. Torque and Direction Analysis (tau = Ialpha): The applied force F at radius R_1 produces a torque about the center of mass. We analyze the sense of rotation caused by this torque. The moment of inertia I of the spool is calculated using its mass (M) and radius of gyration (K = 4\text{ cm}) via the relation I = MK^2. By combining Newton's second law for translation (F_{\text{net}} = Ma) and rotation (\tau_{\text{net}} = I\alpha), we set up the dynamic equations of motion. Condition for Pure Rolling (No Slipping): For rolling without slipping, the kinematic constraint relating linear acceleration (a) and angular acceleration (alpha) is given by a = alpha/ R_2. Using this condition, we solve simultaneously for the linear acceleration (a) and the required friction force #Physics #RotationalMotion #Mechanics #Torque #RollingWithoutSlipping #JEEPhysics #PhysicsProblems #EngineeringMechanics #SpoolProblem #NikunjMadhav #PhysicsForJEEAdvanced #JEEAdvanced #JEE2025 #PhysicsPreparation #IITJEE #JEEAspirants #PhysicsLecture #JEEMotivation #IITPreparation #StudyWithMe #JEEExam #Education #Coaching #JEEPhysics