Euler’s Laws and Rigid Bodies

Euler generalized Newton’s laws to systems of particles, including rigid bodies. A rigid body B may be regarded kinematically as a reference frame. It is easy to show that the position of every point in B is determined in a frame of reference F if (a) the position of any point fixed in B, such as its mass center C, is known in the frame of reference F; and (b) the orientation of B is known in F.

Euler’s first law for a rigid body simply states that the resultant force acting on a rigid body is equal to its mass times the acceleration of the body’s mass center in an inertial frame. Euler’s second law is more involved and may be stated in several ways. The two ways used most commonly in this text are as follows:

• The sum of torques about the mass center C of a rigid body is equal to the time rate of change in F of the body’s angular momentum in F about C, with F being an inertial frame.

• The sum of torques about a point O that is fixed in the body and is also inertially fixed is equal to the time rate of change in F of the body’s angular momentum in F about O, with F being an inertial frame. We subsequently refer to O as a “pivot.”

Consider a rigid body undergoing two-dimensional motion such that the mass center C moves in the x-y plane and the body has rotational motion about the z axis. Assuming the body to be “balanced” in that the products of inertia Ixz = Iyz = 0, Euler’s second law can be simplified to the scalar equation

Tc = Ic в (2.4)

where Tc is the moment of all forces about the z axis passing through C, IC is the moment of inertia about C, and в is the angular acceleration in an inertial frame of the body about z. This equation also holds if C is replaced by O.