Theoretical part. Physical pendulum – is the solid, which can rotate relatively to the arbitrary of the horizontal axis that doesn’t pass through the center of mass

Physical pendulum – is the solid, which can rotate relatively to the arbitrary of the horizontal axis that doesn’t pass through the center of mass. Moment of the gravity force mg, the arm of which is equal L·sin α. Value of L is the distance from pivot О (suspension center) to point С – the center of the mass of the body. Under the action of this moment the body turns round the suspension center О.

Figure 6.2

Write down the fundamental equation of the rotational motion dynamics

, (6.1)

where I is the moment of inertia of the body, is angular acceleration, minus accounts that the moment of force of mg reduces the angle α. Thus, we get the differential equation of physical pendulum free oscillations

. (6.2)

If angle α is small (less than 5^о) we can consider that sin α = α. We get

. (6.3) Comparing the received equation with the general equation of free harmonic oscillations

, (6.4) where - is cyclic frequency of oscillations, Т – period. Let’s get the equation of the period of oscillations

. (6.5) Equation solution (6.4) is the harmonic function which is the equation of free harmonic oscillations

. (6.6) For performing the first task point we need to change the moment of inertia of the pendulum. It is carried out by moving the weight 8 along the rod 7. But in this way the mass center position changes, it is the distance L.

Figure 6.3

The moment of inertia of the pendulum relatively to the point О is equal to the amount of moment of inertia of the load and rod.

Taking into account Steiner theorem, we obtain

.

Thus, the moment of inertia of the pendulum as function of distance Z from point of suspension to weight center

. (6.7) Let’s find the position of point С the mass center of pendulum which is the distance L as function Z. By the law of moments relatively to the point C we have:

. (6.8)

From figure 3 we can see that

, . (6.9) From equations (6.8) – (6.9) we get

. (6.10) Substitution of (6.7) and (6.10) into (6.5) after squaring (6.5) gives us

, (6.11)

where

,

,

а

. (6.12)

Thus, dependence Y = f(X) according to the theory, must be linear. Experimentally researched is the dependence of oscillations period Т of physical pendulum and distance Z from weight to suspension center. We plot the graph (6.12)

Y=f(X).

If you get a rectilinear graph, it confirms validity of theoretical formulas (6.5) and (6.7), and on its slope we can find the free fall acceleration g. Its coincidence with tabulated value 9,8 m/s² confirms truth of the theoretical ratio.