Kinetic Energy for a Rigid Body

Kinetic energy for a rigid body in planar motion is calculated using the relative velocity of each particle with respect to an arbitrary point A. The position vector extends from point A to each mass element. Using the scalar product and expressing the equation in integral form with a vector identity, the total kinetic energy is expressed in terms of angular momentum about that reference point.

When the reference point A coincides with the center of mass G, the integral of the position vector and mass element equals zero. This simplifies the kinetic energy into two distinct components: the kinetic energy of the center of mass itself and the rotational kinetic energy of the body about its center of mass.

When point A is fixed on the rigid body, the kinetic energy equation becomes simplified compared to an arbitrary reference point. Using the definition of angular momentum of the rigid body, the kinetic energy equation can be expressed in component form, making calculations more manageable for fixed-point analysis.

The position vector rA extends from the reference point A to each mass element i of the rigid body. This vector is essential for defining the relative velocity of each particle and is used in the scalar product to calculate kinetic energy. The integral of this position vector with mass elements determines whether the kinetic energy expression simplifies.

The scalar product is used to combine the relative velocity vectors with mass elements when calculating kinetic energy for each particle. This mathematical operation, combined with integral form and vector identities, transforms the individual particle kinetic energies into a unified expression for the entire rigid body.

The kinetic energy of a rigid body referenced to its center of mass consists of translational kinetic energy, which depends on the velocity of the center of mass, and rotational kinetic energy, which depends on the angular velocity about the center of mass. Together, these components fully describe the total kinetic energy of the moving rigid body.

The vector identity is applied after expressing the kinetic energy equation in integral form. This mathematical tool helps transform the complex expression involving relative velocities and position vectors into a more manageable form that can be expressed in terms of angular momentum and moment of inertia properties of the rigid body.