Katerina Tokovenko

The moment of inertia of a body was deduced by considering the rotation of the rigid body. But it is characteristic not only of the rotating body as each body has its own moment of inertia independently of whether it is rotated or in a solid-state. A body has the minimum value of the moment of the inertia when the axis of rotation goes through the center of mass. The moment of inertia with respect to an axis that goes through the center of mass is called the principal moment of inertia.

Let us calculate the moment of inertia in certain cases of continuous distribution of the volumetric mass.

For the entire body, the Formula J = ∑ m_ir_i² is approximate. If the body is divided into a great number of small pieces dm, their addition comes to the integration operation:

J = ∫ r²dm

The distribution of volumetric mass is characterized by density

Whence it follows that dm = ρ dv.

In general ρ may be changed into the volume of a body, i.e. ρ = ρ (r).

Thus, finding of the moment of inertia of a body comes to calculation of integral

J = ∫ ρ (r) r²dv.

1. Let us calculate the moment of inertia of the entire disc with the mass m and radius R about an axis z that goes through the center of the disc (fig. 3.4). Let us divide the disc into rings with the width dr. The volume of such layer is

Dv = 2π rhdr,

Where h is the width.

The mass of the layer is

Dm = ρ dv = ρ 2π rhdr.

Since the density of all points is equal and it is possible to take ρ outside the integral sign, it is derived that

If the mass of disk m = ρ π hR² is taken into consideration, the formula for the moment of inertia may be illustrated in a such written form

The formula (3.14) defines the moment of inertia of the disk about the axis that goes through the center of mass.

2. Let us calculate the moment of inertia of a core with the mass m and the length l about the axis that goes through its center of mass (fig 3.5). Let us divide the core into small pieces dm that are located at a distance x. The volume of every such component is dv = dx · S, where S is a plane of section; the mass is dm = ρ dv = ρ Sdx.

The moment of inertia of the entire core is

The mass of the core is m = ρ Sl, so J₀ – the moment of inertia about the center of mass,

3. Let an axis of rotation z of a core goes through some end of a core (fig. 3.6):

That is

Thus, if the axis of rotation is transferred in such a way that the distance from the axis of rotation r_i is increased, the moment of inertia is also increased. The calculation of the moment of inertia of a body that has a complex shape is a quite complicated problem but it may be partly facilitated if the Huygens-Steiner theorem that is about the parallel axis is used. It defines that if the moment of inertia J₀ of a body that has the mass m about the axis that goes through the center of mass of this body is known, the moment of inertia about another axis that is parallel to the first and is located at the distance a is

J_z = J₀ + ma². (3.15)

Let us consider an arbitrary-shaped body for the Huygens-Steiner theorem proving (fig. 3.7). Let us take two parallel axis one of which goes through the center of mass of the body O and another one through the point O`. Axes are perpendicular to the plane of a figure. Let us draw the coordinate system x, y, z in such a way that its zero point is in the center of mass of the body O, and the axis z lies in the axis O. The coordinate system x`, y`, z` must be located in such a way that the axis z lies in the axis O` and the zero point must be located in such a way that the axes x and x` are lied in. Coordinates of the material point are Δ m_i in both coordinate systems (x_i, y_i,) and (a+x_i, y_i).

The moment of inertia about axis O is

The moment of inertia about axis O` is

Since the zero point of x, y, z lies in the center of mass of the body so coordinate of the center of mass x_C equals to zero and then the third summand of expression is missing. Consequently, the expression acquires such form:

or

J_z = J₀ + ma²,

where