Physical pendulum

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A physical pendulum (also called a physical pendulum or inertial pendulum ) is a theoretical model that describes the oscillation of a real pendulum. In contrast to the mathematical pendulum , the shape and size of the body are taken into account, which means that the behavior of physical pendulums corresponds more closely to the real pendulum .

The physical pendulum consists of an extended, rigid body that is not suspended in its center of gravity . If you move it out of its equilibrium position, it begins to vibrate under the influence of gravity . Frictional force and larger amplitudes are not taken into account in favor of solvability .

The period of oscillation of the physical pendulum results in

where is the angular frequency , the moment of inertia with respect to the point of suspension, the mass of the body, the acceleration due to gravity and the distance from the point of suspension to the center of mass.

One application of the physical pendulum is the experimental determination of the moment of inertia.

Reduced pendulum length

The reduced pendulum length means the length , equivalent to the length in the oscillation equation of the mathematical pendulum with the same oscillation period. At the same time is this size of vibration or shock midpoint set. This location, which cannot be confused with the center of gravity of the pendulum, has the property that an impact directed there does not generate any bearing reaction at the point of suspension of the pendulum. Furthermore, the period of oscillation of a physical pendulum does not change if the point of suspension and center of oscillation are swapped (see also reversion pendulum ).

Mathematical description

To calculate the period of oscillation, two different approaches are used for the torque acting on the physical pendulum, and , which can also be applied to the mathematical pendulum.

Assume that the pendulum body is suspended at the origin and can swing in the xy-plane , whereby gravity acts in the negative y-direction. Then the position of the body's center of gravity can be described by in the resting state and by in the deflected state . The torque acting on the physical pendulum can now be calculated as for a point mass of the same mass located in the center of gravity of the pendulum:

The torque acting on the pendulum has only one component in the z-direction, so it is perpendicular to the plane of oscillation.

By equating it with the approach (torque of an extended body) and subsequent reshaping, the equation is obtained , whereby the sine for small angles can also be approximated . The equation describes a harmonic oscillation with the period of oscillation of the pendulum .