Theoretical part. A physical pendulum is a body which can be revolved relatively of arbitrary horizontal axis, that not go through the center of mass

A physical pendulum is a body which can be revolved relatively of arbitrary horizontal axis, that not go through the center of mass. Under the action of moment of force mg, the arm of which is L·sinα, a body is revolved round the point of hang up O (fig.10.2). L is distance from a point O rotation (points of hang up) to the point of C - center of mass of body. Write down the fundamental equation of the rotational motion dynamics

, (10.1) I is a moment of inertia of body, is angular acceleration. A sign does minus take into account, that the moment of force of mg is diminished by a corner α.

Figure 10.2

Thus, get differential equation of undamped oscillation of the physical pendulum

. (10.2)

At small corners α (less 5^о) is it possible, that sin α = α. Get

(10.3) Comparing this equation to general equation of undamped harmonic oscillations

, (10.4)

get cyclic frequency and period of oscillation of the physical pendulum

(10.5)

Thus, the period of oscillation of the physical pendulum depends on position of point of hang up O and forms of body, that to its moment of inertia in relation to this point.

For a mathematical pendulum, which is a material point, suspended on a weightless unstretching thread long L, moment of inertia is , . Consequently the period of oscillation of the mathematical pendulum depends only on length of thread

. (10.6)

The resulted length of L_res of physical pendulum is such length of mathematical pendulum the period oscillation of which equals the period of oscillation of the physical pendulum. From (10.5) and (10.6) we have

. (10.7)

A moment of inertia of peg (fig.10.1) is taking into account a theorem Steiner

. (10.8)

Thus, from (10.7) and (10.8) get the theoretical value of the resulted length

. (10.9)

Find the theoretical value of period of oscillation of the physical pendulum from (10.5) and (10.8)

. (10.10)