You get a bonus - 1 coin for daily activity. Now you have 1 coin

5.4. Moment of inertia of a solid

Lecture



To find the moment of inertia of the body, it is necessary to sum the moment of inertia of all material points that make up the body.

  5.4.  Moment of inertia of a solid (5.4)

In general, if the body is solid, it is a collection of points with infinitely small masses.   5.4.  Moment of inertia of a solid , and the moments of inertia of the body is determined by the integral

  5.4.  Moment of inertia of a solid (5.5)

about where   5.4.  Moment of inertia of a solid - distance from the element   5.4.  Moment of inertia of a solid to the axis of rotation.

The distribution of mass within the body can be characterized by
density

  5.4.  Moment of inertia of a solid (5.5)

where m is the mass of a homogeneous body, V is its volume. For a body with an unevenly distributed mass, this expression gives an average density.

  5.4.  Moment of inertia of a solid

The density at this point in this case is determined as follows.

  5.4.  Moment of inertia of a solid

and then

  5.4.  Moment of inertia of a solid (5.6)

The limits of integration depend on the shape and size of the body. Integration of equation (5.5) is most easily implemented for those cases when the axis of rotation passes through the center of gravity of the body. Let us consider the results of integration for the simplest (geometrically regular) forms of a solid body whose mass is uniformly distributed over the volume.

  5.4.  Moment of inertia of a solid

The moment of inertia of a hollow cylinder with thin walls, radius R.

For thin-wall hollow cylinder

  5.4.  Moment of inertia of a solid

Solid homogeneous disk. The axis of rotation is the axis of the disk radius   5.4.  Moment of inertia of a solid . and mass m with density   5.4.  Moment of inertia of a solid Disk height h. Cut a hollow cylinder with a wall thickness inside the disk at a distance   5.4.  Moment of inertia of a solid and mass   5.4.  Moment of inertia of a solid . For him

  5.4.  Moment of inertia of a solid

The entire disk can be divided into an infinite number of cylinders, and then summed:

  5.4.  Moment of inertia of a solid

The moment of inertia of the ball relative to the axis passing through the center of gravity.

  5.4.  Moment of inertia of a solid

The moment of inertia of a rod of length L and mass m relative to the axis passing:

a) through the center of the rod -   5.4.  Moment of inertia of a solid

b) through the beginning of the rod -   5.4.  Moment of inertia of a solid

  5.4.  Moment of inertia of a solid

Steiner theorem. We have a body, the moment of inertia of which is relative to the axis passing through its center of mass   5.4.  Moment of inertia of a solid known. It is necessary to determine the moment of inertia about an arbitrary axis.   5.4.  Moment of inertia of a solid parallel to the axis   5.4.  Moment of inertia of a solid . According to Steiner’s theorem, the moment of inertia of a body about an arbitrary axis is equal to the sum of the moment of inertia of a body about an axis passing through the center of mass and parallel to this axis, plus the product of the body mass per square of the distance between the axes:

  5.4.  Moment of inertia of a solid (5.7)


Comments


To leave a comment
If you have any suggestion, idea, thanks or comment, feel free to write. We really value feedback and are glad to hear your opinion.
To reply

Physical foundations of mechanics

Terms: Physical foundations of mechanics