# Rigid Body

A rigid body is a solid body made up of particles that always remain at the same distance from each other regardless of the forces that act on it. In other words, a rigid solid (or body) does not undergo any type of deformation. The motion of rigid body can be very complex but, by making the necessary decompositions, we can approach its study in a simplified way.

The fundamental difference with respect to Newtonian dynamics is that a rigid body has volume, so it can have a rotational motion.

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The study of the motion of a rigid body can be approached in two ways: by analyzing the forces acting on it or using the law of conservation of energy.

Newton’s second law describes the translational motion of a particle, but does not describe the rotational motion. For this we need a new equation, which is sometimes called Newton’s second law for rotation, and which is given by: The first member of the above equation is the resultant torque (or moment) of the external forces acting on the solid. This magnitude is defined: Where r is a vector that goes from the point we choose to calculate the torques (which will depend on each particular situation) to the point of application of the force F.

It is important to highlight that, contrary to what happened with the translational motion of a particle, the rotational motion of a solid depends on the application point of the forces acting on it.

In the second member of Newton’s second law for rotation I is the moment of inertia of the solid and α the angular acceleration of the same.

The angular acceleration of a rigid body is always parallel to that resulting from the torques of external forces acting on it.

The moment of inertia is the analogue to the mass of a body in the rotational motion. For a rigid body it is defined: Where k is a constant and R a characteristic length of the solid (typically its radius or its length). From the above definition it follows that the moment of inertia depends on the axis around which the rigid body rotates.

In these pages we will see some examples of application of these equations.

When a body is rotating, its kinetic energy depends on its angular velocity ω and is given by: As you will see in the examples discussed in these pages, this term will have to be included in the energy of a rigid body when we will study its movement applying the law of conservation of energy.

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