How does the center of gravity affect balance?
Torque and balance¶
The moment of inertia¶
If an object is tilted so far by an external torque that its equilibrium becomes unstable, it begins to rotate.
In the case of translational movements, the acceleration is a body created by an external force experiences, inversely proportional to the mass of the body ("law of force"). The angular acceleration is corresponding with rotational movementsthat a body is caused by an external torque experiences, inversely proportional to the so-called moment of inertia of the body.
So the following applies:
For translational movements:
For rotational movements:
The torque points in the same direction as the angular acceleration, i.e. points in the direction of the axis of rotation; the magnitude of the torque vector is by the factor the amount of the angular acceleration different.
The value of the moment of inertia of a body depends on the one hand on the position of the axis of rotation and on the other hand on the spatial distribution of the rotating mass. Masses that are far from the axis of rotation contribute more to the moment of inertia than masses that are near the axis of rotation. For example, a small particle has mass that is on a circular path with a radius rotates, a moment of inertia of .
The moment of inertia of a body of any shape can be determined mathematically if it is made up of a large number of individual small pieces of mass put together thinks each at a distance away from the axis of rotation. The body's moment of inertia is then equal to the sum of the moments of inertia of all individual parts:
The moments of inertia of some regularly shaped bodies, which often occur in technical applications, are listed in formulas. A small selection of these can be found in Figure Moments of Inertia. It should be noted that the same body can have different moments of inertia with respect to different axes of rotation.
Rotates a body with a mass around an axis that is in distance runs parallel to the center of gravity, it must be at the respective value the term of the moment of inertia can be added. This can be explained by the fact that in this case the center of gravity of the body rotates on a circular path around the new axis of rotation. The following applies:
Formally, the new moment of inertia is therefore equal to the original moment of inertia (when rotating around the center of gravity) plus the moment of inertia of the center of gravity around the new axis of rotation. In this way, many rotational processes (in particular rolling movements) can be attributed in a simple manner to rotational movements around the center of gravity.
Experimental determination of the moment of inertia
In order to experimentally determine the moment of inertia of any object, for example a rotating disk or a wheel, it can be hung freely rotatable, for example by means of a holder above its center of gravity.
If this arrangement is deflected from the rest position, it begins as a physical pendulum with an easily measurable period of oscillation to swing the rest position. The following then applies to the moment of inertia of the object around its center of gravity:
Here referred to the acceleration due to gravity, the mass of the vibrating object, and the distance between its center of gravity and the suspension point. Since all other quantities in the above formula are either constant or also easily measurable, the moment of inertia can be determined directly.
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