1. Introduction to Rigid Body Motion

Rotational motion describes the movement of an object around a fixed axis or point. A rigid body is an idealized object where the distance between any two constituent particles remains constant, meaning it does not deform under applied forces. Analyzing rigid body motion involves understanding both the translation of its center of mass and its rotation around that center. This branch of physics extends the concepts of linear kinematics and dynamics to include angular displacement, angular velocity, and angular acceleration, allowing us to describe complex movements like the spinning of a top or the rotation of a wheel.

2. Center of Mass and Linear Momentum of Systems

The center of mass (CM) is a point that represents the average position of all the mass in a system. For a system of particles, its motion can often be described as if all the mass were concentrated at the CM and all external forces acted there. The linear momentum of a system of particles is the vector sum of the momenta of all individual particles. The total linear momentum of a system is conserved if no net external force acts on it, meaning the CM of the system moves with constant velocity. This principle is fundamental in analyzing collisions and the behavior of multi-particle systems.

3. Vector Products and Angular Quantities

Rotational motion inherently involves vector quantities, and their manipulation often requires the use of vector products. Angular displacement, angular velocity ($\vec{\omega}$), and angular acceleration ($\vec{\alpha}$) are all vectors. The relationship between linear velocity ($\vec{v}$) and angular velocity is given by $\vec{v} = \vec{\omega} \times \vec{r}$, where $\vec{r}$ is the position vector from the axis of rotation. Similarly, angular momentum ($\vec{L}$) is a vector product of the position vector and linear momentum ($\vec{L} = \vec{r} \times \vec{p}$). The torque ($\vec{\tau}$), the rotational analogue of force, is also a vector product: $\vec{\tau} = \vec{r} \times \vec{F}$.

4. Equilibrium of a Rigid Body and Center of Gravity

A rigid body is in rotational equilibrium if the net torque acting on it is zero, and in translational equilibrium if the net force acting on it is zero. For an object to be in complete equilibrium, both conditions must be met. The center of gravity (CG) is the point where the entire weight of the body can be considered to act. For a uniform gravitational field, the CG coincides with the center of mass. Understanding the CG is crucial for analyzing stability, such as how an object balances on a pivot point.

5. Moment of Inertia and its Theorems

The moment of inertia ($I$) is the rotational analogue of mass; it is a measure of an object's resistance to changes in its rotational motion. It depends on the mass of the object and how that mass is distributed relative to the axis of rotation. For a point mass $m$ at a distance $r$ from the axis, $I = mr^2$. For a rigid body, $I = \sum mr^2$. The Parallel Axis Theorem ($I = I_{CM} + Md^2$) and the Perpendicular Axis Theorem ($I_z = I_x + I_y$ for a planar object) are essential theorems that simplify the calculation of moments of inertia for complex shapes by relating them to the moment of inertia about the center of mass.

6. Dynamics of Rotational Motion and Rolling

The rotational analogue of Newton's second law is $\vec{\tau}_{\text{net}} = I\vec{\alpha}$, where $\vec{\tau}_{\text{net}}$ is the net external torque and $\vec{\alpha}$ is the angular acceleration. This equation governs the dynamics of rotational motion, explaining how torques cause objects to rotate. Rolling motion, a combination of translation and rotation (e.g., a wheel rolling down a hill), can be analyzed by considering both the linear motion of the center of mass and the rotational motion. The condition for pure rolling is that the velocity of the point of contact with the surface is zero, relating linear and angular velocities ($v_{CM} = R\omega$).

7. Additional: Work, Energy, and Power in Rotation

Similar to linear motion, rotational motion also involves work, energy, and power. The rotational work done by a torque $\tau$ over an angular displacement $\Delta \theta$ is $W_{\text{rotational}} = \tau \Delta \theta$. The rotational kinetic energy of a body is given by $KE_{\text{rotational}} = \frac{1}{2}I\omega^2$. The Work-Energy theorem applies here as well: the net work done on a rotating object equals its change in rotational kinetic energy. Rotational power is the rate at which rotational work is done, $P_{\text{rotational}} = \tau \omega$, analogous to $P = Fv$ in linear motion. These concepts are vital for understanding the energy transformations in rotating systems.