In 1673 Huygens had shown that the period of a rigid bar pendulum ( called a " compound pendulum " ) was equal to the period of a simple pendulum with a length equal to the distance between the center of oscillation, located under the center of gravity, that depends on the mass distribution along the pendulum.
22.
In the swinging of a simple pendulum, Galileo says in " Discourses " that " every momentum acquired in the descent along an arc is equal to that which causes the same moving body to ascend through the same arc . " His analysis on projectiles indicates that Galileo had grasped the first law and the second law of motion.
23.
As long as each pivot is close to the center of oscillation of the other, so the two periods are close, the period " T " of the equivalent simple pendulum can be calculated with equation ( 2 ), and the gravity can be calculated from " T " and " L " with ( 1 ).
24.
If a pendulum was built with a second adjustable pivot point near the bottom so it could be hung upside down, and the second pivot was adjusted until the periods when hung from both pivots were the same, the second pivot would be at the center of oscillation, and the distance between the two pivots would be the length " L " of a simple pendulum with the same period.
25.
Suppose we have a bead sliding around on a wire, or a swinging simple pendulum, etc . If one tracks each of the massive objects ( bead, pendulum bob, etc . ) as a particle, calculation of the motion of the particle using Newtonian mechanics would require solving for the time-varying constraint force required to keep the particle in the constrained motion ( reaction force exerted by the wire on the bead, or tension in the pendulum rod ).
26.
By considering limit cases, the correctness of this system can be verified : For example, \ ddot x \ to 0 should give the equations of motion for a simple pendulum that is at rest in some inertial frame, while \ ddot \ theta \ to 0 should give the equations for a pendulum in a constantly accelerating system, etc . Furthermore, it is trivial to obtain the results numerically, given suitable starting conditions and a chosen time step, by stepping through the results iteratively.
