CHAPTER V.

MOMENT OF INERTIA.

THE Moment of Inertia of a body, with regard to any axis, is the sum of all the products resulting from the multiplication of each element of the mass by the square of its distance from the axis. If M denote the whole mass of the body, the Moment of Inertia may be represented by the expression M3, where k is a line called the Radius of Gyration. The term Moment of Inertia was first made use of by Euler. nationis ex similitudine motus progressivi est desumpta: quemadmodum enim in motu progressivo, si à vi secundum suam directionem sollicitante acceleretur, est incrementum celeritatis ut vis sollicitans divisa per massam seu inertiam; ita in motu gyratorio, quoniam loco ipsius vis solicitantis ejus momentum

"Ratio hujus denomi

considerari oportet, eam expressionem frdM, quæ loco inertiæ

in calculum ingreditur, momentum inertiæ appellemus, ut incrementum celeritatis angularis simili modo proportionale fiat momento vis sollicitantis diviso per momentum inertiæ""

SECT. 1. A Plane Curve about an Axis within its Plane.

(1) To find the moment of inertia and radius of gyration of a circular arc about a radius through its vertex.

Let HAK (fig. 174) be the circular arc, A its vertex, C the centre of the circle. Take any point P in the arc; draw PM at right angles to the radius CA; join HK, intersecting CA in E. Join CH, CK, CP. Let PM-y, CA = a, arc AP=s, HE =c=RE, ACH=α=ACK, ▲ PCA = 0. Then, the density of the arc and the indefinitely small area of the section of it

1 Euler; Theoria Motus Corporum Solidorum, p. 167.

made by a plane through C, at right angles to its own plane, being represented respectively by p and x, we shall have

If the arc be a semicircle, c = a, and, if a circle, c = 0; in both cases k2 = {a2.

(2) To find the radius of gyration of a material straight line OB, (fig. 175), about an axis OA, to which it is inclined at a given angle, the density at any point of OB varying as some power of its distance from O.

Take any point P in OB; draw PM at right angles to OA; let PM=y, OP=s, OB=l, ▲ AOB = a, p= the density at P: then p = μs", where μ is a constant quantity. Hence

If the density be invariable, n = 0, and k2 = 37 sin2 a.

SECT. 2. A Plane Carve about an Axis at Right Angles to its Plane.

(1) To find the radius of gyration of a straight line AB, (fig. 176) about an axis through D at right angles to the plane ADB.

Let C be the middle point of the line; join CD. Let AC=a =BC, CD=b; k = the radius of gyration about the axis through D, and k' that about an axis parallel to this through C. Then k2 = k'2 + b2.

=

(2) To find the radius of gyration of a circular arc about an axis perpendicular to its plane through its centre of gravity.

Let k be the radius of gyration about the required axis, k' about an axis parallel to this through the centre of the circle, and h the distance between the centre of gravity of the arc and the centre of the circle. Then k'2 = k2 + h2.

But, r denoting the radius of the circle, c the chord, and a the length of the arc,

(3) To find the radius of gyration of a circular arc about an axis perpendicular to its plane through its vertex.

If r = the radius of the circle, a = the length and c = the chord

(4) If the density of a straight rod AB vary as the nth power of the distance from one end A, and k, k', be the radii of gyration of the rod round axes at right angles to its length through A and B respectively; to compare the values of k and k', and to ascertain the value of n so that k may be equal to 6k'.

(5) To find the shape of a uniform wire, lying in one plane, such that the moment of inertia of any portion about an axis perpendicular to its plane may vary as the difference of the distances of its extremities from the axis.

The equation to the form of the wire, a and a being constants, and the point where the axis intersects the plane being the origin of polar co-ordinates, is

SECT. 3. A Plane Area about an Axis within or parallel to its Plane.

(1) To find the radius of gyration of an elliptic area, of uniform thickness and density, about its principal axes.

Let p represent the uniform density of the area, and ▾ its indefinitely small thickness; then, x, y, denoting the co-ordinates of any point of the curve referred to the axes of the ellipse as axes of co-ordinates, we have for the moment of inertia, about the major axis of a quadrant of the ellipse,

Hence the moment of inertia of the whole ellipse will be equal

If ' denote the radius of gyration about the minor axis, we shall have, by similar reasoning,

k"2 = {a2.

(2) To find the radius of gyration of a circular area about a straight line parallel to its plane, at a distance c from its

centre.

If a be the radius of the circle, and k the required radius of gyration,

k2 = {a2 + c2.

(3) To find the radius of gyration of an isosceles triangle about a perpendicular let fall from its vertex upon its base.

(4) To find the radius of gyration of a lamina, bounded by the lemniscate r2 = a2 cos 20, about the axis of the curve.