### Introductioncenter Of Masssystem Of Particles And Rotational Motion

## Concepts to remember for the : Introduction - Center of Mass - System of Particles, and Rotational Motion

**Centre of mass**: The center of mass of a system of particles is the unique point where the total mass of the system can be considered to be concentrated. It is the point at which the net external force on the system would act if the system were considered to be a single particle.

**Position vector of the center of mass**: The position vector of the center of mass of a system of particles is given by the formula,
$$ \vec{r_{cm}} = \frac{\sum_i m_i \vec{r}_i}{M} $$
where,

- (m_i) is the mass of the i-th particle
- (\vec{r}_i) is the position vector of the i-th particle
- (M = \sum_i m_i) is the total mass of the system

**Center of mass of a two-particle system**: The center of mass of a two-particle system lies on the line joining the two particles and divides the distance between them in inverse proportion to their masses.

**Center of mass of a system of particles**: The center of mass of a system of particles is the point at which the total mass of the system can be considered to be concentrated. It is the point at which the net external force on the system would act if the system were considered to be a single particle.

**Translational motion**: Translational motion is the motion of a rigid body in which all points in the body move in parallel lines with equal speeds.

**Rotational motion**: Rotational motion is the motion of a rigid body about a fixed axis.

**Angular displacement**: The angular displacement of an object is the measure of the amount of rotation it has undergone about a fixed axis. It is measured in radians.

**Angular velocity**: The angular velocity of an object is the rate of change of its angular displacement. It is measured in radians per second.

**Angular acceleration**: The angular acceleration of an object is the rate of change of its angular velocity. It is measured in radians per second squared.

**Torque**: Torque is the turning effect of a force acting on a body about a fixed axis. It is measured in newton-meters.

**Moment of inertia**: The moment of inertia of an object is the measure of its resistance to rotational motion. It depends on the mass distribution of the object and the axis of rotation. It is measured in kilogram-meters squared.

**Parallel axis theorem**: The parallel axis theorem states that the moment of inertia of an object about an axis parallel to its center of mass is equal to its moment of inertia about the center of mass plus the product of its mass and the square of the distance between the two axes.

**Perpendicular axis theorem**: The perpendicular axis theorem states that the moment of inertia of an object about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two mutually perpendicular axes in the plane.

**Kinetic energy of rotation**: The kinetic energy of rotation of an object is the energy it possesses due to its rotation. It is given by the formula, $$K_r=\frac{1}{2} I \omega^2 $$

**Work done in rotational motion**: The work done in rotational motion is the energy transferred to an object due to its rotation. It is given by the formula, $$W=\tau \theta$$

**Power in rotational motion**: The power in rotational motion is the rate at which work is done in rotational motion. It is given by the formula, $$P=\tau \omega $$

**Rolling motion**: Rolling motion is the motion of an object that is in contact with a surface and rotates about an axis that is fixed to the surface.

**Conservation of angular momentum**: The conservation of angular momentum states that the total angular momentum of a system remains constant in the absence of an external torque.