This topic explores the collective behavior of systems composed of multiple particles, focusing on their translational and rotational motion. It bridges the gap between the motion of individual particles and the motion of rigid bodies, providing a foundation for understanding a wide range of phenomena, from the motion of galaxies to the spinning of a top.
1. System of Particles
- Center of Mass: A point that represents the average position of all the mass in a system. It moves as if the entire mass of the system were concentrated at that point.
- Motion of Center of Mass: The center of mass of a system moves as if it were a single particle acted upon by the net external force on the system.
- If the net external force is zero, the center of mass moves with constant velocity or remains at rest.
- Linear Momentum of a System: The vector sum of the linear momenta of all the particles in the system. It is equal to the product of the total mass of the system and the velocity of its center of mass.
- Conservation of Linear Momentum: If the net external force on a system is zero, the total linear momentum of the system remains constant.
2. Rotational Motion
- Rigid Body: A body in which the relative positions of all its particles remain constant.
- Rotation about a Fixed Axis: Motion where every particle of the rigid body moves in a circle about a fixed line called the axis of rotation.
- Angular Velocity (ω): The rate of change of angular displacement. It is a vector quantity with direction along the axis of rotation.
- Angular Acceleration (α): The rate of change of angular velocity.
- Torque (τ): The rotational equivalent of force. It is the tendency of a force to cause rotation. Torque is equal to the cross product of the force vector and the position vector from the axis of rotation to the point of application of the force.
- Moment of Inertia (I): The rotational equivalent of mass. It is a measure of a body’s resistance to angular acceleration. It depends on the mass distribution of the body and the axis of rotation.
- Angular Momentum (L): The rotational equivalent of linear momentum. It is equal to the product of the moment of inertia and the angular velocity.
- Conservation of Angular Momentum: If the net external torque on a system is zero, the total angular momentum of the system remains constant.
3. Rolling Motion
- Combination of Translation and Rotation: Rolling motion is a combination of translational motion of the center of mass and rotational motion about the center of mass.
- Pure Rolling: Rolling without slipping, where the point of contact with the surface is instantaneously at rest.
- Kinetic Energy of Rolling: The total kinetic energy of a rolling body is the sum of its translational kinetic energy and its rotational kinetic energy.
4. Important Theorems
- Parallel Axis Theorem: Relates the moment of inertia of a body about an axis to its moment of inertia about a parallel axis through its center of mass.
- Perpendicular Axis Theorem: Relates the moments of inertia of a planar body about three mutually perpendicular axes.
Applications
- Gyroscopes: Devices used for navigation and stabilization, based on the principle of conservation of angular momentum.
- Flywheels: Used to store rotational kinetic energy and smooth out fluctuations in rotational speed.
- Rotating machinery: Understanding rotational motion is crucial for designing and analyzing engines, turbines, and other rotating machinery.
- Celestial mechanics: The principles of rotational motion are essential for understanding the motion of planets, stars, and galaxies.