## Planar Motion of a Slender

In this section, we consider the planar motion of a thin and slender rod with uniformly distributed mass. We explore the motion of its center of mass as well as its rotation. The mechanical analysis of motion of a rod allows us to introduce in elementary form all the basic principles of rigid body mechanics. Also, in many applications, either the human body or various long bones of the extremities can be considered as rigid rods; thus, the geometry chosen has significance to human body dynamics.

Consider the swinging motion of a rod with uniformly distributed mass in a vertical plane as shown in Fig. 4.1. The moment of momentum of the particles in the rod with respect to the hinge O can be written as an integral summation over small mass elements of the body:

where r is the position vector from point O to the center of mass of the mass element dm, and v is the velocity of the center of mass of dm in the reference frame E. The integration is over all the small mass elements of the slender rod (Fig. 4.1). From the geometry of the pendulum movement, we deduce the following equations for the position r and velocity v of dm, which is located at a distance s away from point O:

in which $ is the angle of the rod with the e2 axis as shown in Fig. 4.1. The time derivative of angle $, (d$/dt), is the rate of rotation of the rod. The rod rotates counterclockwise when (d$/dt) is positive and rotates clockwise when (d$/dt) is negative.

The angular velocity w of the rod undergoing planar motion is defined in the vectorial form as follows:

where e3 is the unit vector perpendicular to the plane of motion (Fig.4.1a). Because the angles have no dimension, the unit of angular velocity is inverse time. It is expressed as radians per second (rad/s).

Substituting the position vector and velocity expressions given by Eqns. 4.2a and 4.2b into Eqn. 4.1 and noting that dm = (m/L) ds, we obtain the following relation:

Ho = f{s [sin $ e1 - cos $ e2]} X {(d$/dt) s [cos $ e1 + sin $ e2]} (m/L) ds

After performing the vector multiplications in the equation written above, the moment of momentum expression reduces to

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