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Kinetic Energy of a Body in a Plane

Key Concepts: A rigid body in a plane may both translate and rotate. Consequently its total

kinetic energy must account for both translation of the center of mass and rotation about its

center of mass. Mass and mass moment of inertia enter into the calculation of kinetic energy.

In a Nut Shell: Kinetic energy is a scalar. So add the translational and rotational kinetic energy

directly to obtain the total kinetic energy for plane motion.

By vector addition: r_P = r_C + r_CP = position vector of P at any time t . P is an arbitrary

point in the rigid body, C is at the center of mass of the body. See the figure below.

Now take the derivative of each vector with respect to time in the fixed frame of reference, F.

v_P = v_C + ω x r_CP

The kinetic energy of an element of mass, dm, located at point P is ½ v_P . v_P dm

Integration yields the total kinetic energy, T.

The result for the total kinetic energy of the rigid body, T (for plane motion) is

T = ½ m v_C² + ½ I^C_zz ω²

KE of translation + KE of rotation

where m is the mass of the rigid body concentrated at its center of mass

v_C is the speed of the center of mass of the body measured in the fixed frame of reference

I^C_zz is the mass moment of inertia of the body about its center of mass

ω is the angular speed of the rigid body measured in the fixed frame of reference