System of Particles:
A system of particles refers to a collection of particles, where each particle can move independently or interact with other particles within the system. The behavior of a system of particles can be analyzed using the principles of Newton's laws of motion and the concept of center of mass.

Rotational Motion:
Rotational motion involves the movement of an object around an axis or a fixed point. In this type of motion, the particles of the object move in circular paths about the axis of rotation. The rotational motion can be described using various parameters such as angular displacement, angular velocity, and angular acceleration.

Brain Map:
A brain map, also known as a mind map, is a visual representation of information that helps in organizing and understanding complex concepts. Here is a brain map for the system of particles and rotational motion:

1. System of Particles:
- Definition: Collection of particles interacting with each other.
- Newton's Laws of Motion: Applicable to each particle within the system.
- Center of Mass: The point where the entire mass of the system is considered to be concentrated.

2. Types of Systems:
- Isolated System: No external forces or torques acting on the system.
- Closed System: External forces or torques, but no external particles, acting on the system.
- Open System: External forces, torques, and particles can enter or leave the system.

3. Center of Mass:
- Definition: The average position of all the particles in a system.
- Calculation: Weighted average of individual particle positions.
- Motion: Moves in a straight line if no external forces act on the system.

4. Conservation of Linear Momentum:
- Principle: The total momentum of an isolated system remains constant.
- Equation: Σ(mv) = constant

5. Rotational Motion:
- Definition: Circular motion of particles around an axis or fixed point.
- Parameters: Angular displacement, angular velocity, and angular acceleration.
- Moment of Inertia: Resistance of an object to rotational motion.

6. Torque:
- Definition: The measure of the force's effectiveness in causing rotational motion.
- Calculation: Torque = Force x Perpendicular distance from the axis of rotation.
- Units: Newton-meter (Nm).

7. Conservation of Angular Momentum:
- Principle: The total angular momentum of an isolated system remains constant.
- Equation: Σ(Iω) = constant

8. Rolling Motion:
- Combination of translational and rotational motion.
- Contact point remains at rest during pure rolling.
- Rolling without slipping: v = Rω, where v is the linear velocity, R is the radius, and ω is the angular velocity.

9. Kinetic Energy of Rotation:
- Calculation: KE = (1/2)Iω², where KE is the kinetic energy, I is the moment of inertia, and ω is the angular velocity.

10. Parallel Axis Theorem:
- Relates the moment of inertia about an axis parallel to the original axis.
- Equation: I_parallel = I_cm + md², where I_parallel is the moment of inertia about the parallel axis, I_cm is the moment of inertia about the center of mass, m is the mass of the object