Teach me rotational mechanics as a grade 12 student

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Welcome to rotational mechanics! This is the study of how objects rotate and spin. Just like we studied linear motion with velocity and acceleration, rotational motion has its own set of quantities. We'll explore angular velocity, angular acceleration, torque, and moment of inertia. Let's start by watching this wheel rotate! Angular kinematics describes rotational motion using three key quantities. Angular displacement theta measures how far an object has rotated in radians. Angular velocity omega is the rate of change of angular displacement. Angular acceleration alpha is the rate of change of angular velocity. These are analogous to position, velocity, and acceleration in linear motion. The linear velocity of a point on a rotating object equals the radius times the angular velocity. Torque is the rotational equivalent of force. It's what causes objects to rotate. Torque equals the force times the lever arm times sine of the angle between them. The lever arm is the perpendicular distance from the axis of rotation to the line of action of the force. Newton's second law for rotation states that the net torque equals the moment of inertia times angular acceleration. Just as force causes linear acceleration, torque causes angular acceleration. Rotational kinetic energy equals one half times moment of inertia times angular velocity squared. For objects that both translate and rotate, like a rolling ball, the total kinetic energy is the sum of translational and rotational kinetic energy. Angular momentum equals moment of inertia times angular velocity. Angular momentum is conserved when no external torques act on the system. This is why figure skaters spin faster when they pull their arms in - their moment of inertia decreases, so their angular velocity must increase to conserve angular momentum. Let's see rotational mechanics in action with rolling motion. A rolling sphere combines both translation and rotation. The no-slip condition means the center of mass velocity equals the radius times angular velocity. The total kinetic energy includes both translational and rotational components. You've now mastered the key concepts of rotational mechanics: angular kinematics with theta, omega, and alpha; torque and rotational dynamics; rotational energy and angular momentum; and conservation principles. These concepts are essential for understanding everything from spinning wheels to planetary motion!