O at the equator, goes on increasing till = 45°, where it is a maximum; then goes on decreasing till the latitude is 90° when it again becomes 0.

The effect of the tangential component is to heap up the particles of the earth about the equator, and, were the earth in a fluid state, this process would go on till the effect of the tangential component was exactly counterbalanced by component of gravity acting down the inclined plane thus found, when the particles would be in a state of equili brium. The higher analysis has shown that the form of equilibrium is that of an oblate spheroid, differing but slightly from that which our globe is found to possess by

actual measurement.

From Equation (129), we see that the normal component of the centrifugal force is equal to the centrifugal force at the equator multiplied by the square of the cosine of the latitude of the place.

This component is directly opposed to gravity, and, consequently, tends to diminish the weight of all bodies on the surface of the earth. The value of this component is greatest at the equator, and diminishes towards the poles, where it becomes equal to 0. From the action of the normal component of the centrifugal force, and from the flattened form of the earth due to the tangential component bringing the polar regions nearer the centre of the earth, the measured force of gravity ought to increase in passing from the equator towards the poles. This is found, by observation, to be the case.

The radius of the earth at the equator is found, by measurement, to be about 3962.8 miles, which, multiplied by 2, will give the entire circumference of the equator. If this be divided by the number of seconds in a day, 86400, we find the value of v. Substituting this value of v and that of just given, in Equation (125), we should find,

for the measure of the centrifugal force at the equator. If

this be multiplied by the square of the cosine of the latitude of any place, we shall have the value of the normal component of the centrifugal force at that place.

Centrifugal Force of Extended Masses.

136. We have supposed, in what precedes, the dimensions of the body under consideration to be extremely small; let us next examine the case of a body, of any dimensions whatever, constrained to revolve about a fixed axis, with which it is invariably connected. If we suppose this body to be divided into infinitely small elements, whose directions are parallel to the axis, the centrifugal force of each element will, from what has preceded, be equal to the mass of the element into the square of its velocity, divided by its distance from the axis. If a plane be passed through the centre of gravity of the body, perpendicular to the axis, we may, without impairing the generality of the result, suppose the mass of each element to be concentrated at the point in which this plane cuts the line of direction of the element. Let XCY be the plane through the centre of gravity of the body perpendicular to the axis of revolution, AB the section cut out

Y

0.

m

r

Fig. 121

of the body, or the projection of the body on the plane, and C the point in which it cuts the axis. Take Cas the origin of a system of rectangular axes, and let CX be the axis of X, CY the axis of Y, and let m be the point at which the mass of one of these filaments is concentrated, and denote that mass by m. note the co-ordinates of m by x and y, its distance from C by r, and its velocity by v. The centrifugal force of the mass m will be equal to

mv3
r

De

If we denote the angular velocity of the body by V', the

velocity of the point m will be equal to r", which, being substituted in the expression for the centrifugal force just deduced, gives

mr Vs.

Let this force be resolved into two components, respectively parallel to the axes CX and CY. We shall have, for these components, the expressions,

mr V"cosm CX, and mr Vsinm CX.

Substituting these values in the preceding expressions, and reducing, we have, for the two components,

mx V", and my V".

In like manner, if we denote the masses of the remaining filaments by m', m', &c., the co-ordinates of the points at which they are cut by the plane XCY, by x', y'; x'', y', &c., their distances from the axis by r', r', &c., and resolve the centrifugal forces into components, respectively parallel to the axes, we shall have, since V remains the same,

If we denote the sum of the components in the direction of the axis of X by X, and in the direction of the axis of Y by Y, we shall have,

X =

(mx) V', and Y = Σ(my) V".

If, now, we denote the entire mass of the body, by M, and suppose it concentrated at its centre of gravity 0, whose co-ordinates are designated by x1, and y1, and whose distance from C is equal to r1, we shall have, from the principle of the centre of gravity (Art. 51),

Σ(mx) = Mx1, and (my)

Substituting above, we have,

H

My 1.

X = MV11×19 and Y = MV3y1.

If we denote the resultant of all the centrifugal forces, which will be the centrifugal force of the body, by R, we shall have,

R =√X2 + Y2 = MV"√∞22 + y12 =

But if the velocity of the centre of gravity be denoted by V, we shall have,

which, substituted in the preceding result, gives, for the resultant,

The line of direction of R is made known by the equa

it, therefore, passes through the centre of gravity O. Hence, we conclude, that the centrifugal force of an extended mass, constrained to revolve about a fixed axis, with which it is invariably connected, is the same as though the entire mass were concentrated at its centre of gravity.

Pressure on the Axis.

137. The centrifugal force, passing through the centre of gravity and intersecting the axis, will exert its entire effect in creating a pressure upon the axis of revolution. By inspecting the equation,

R = MV"r1 19

we see that this pressure will increase with the mass, the angular velocity, and the distance of the centre of gravity from the axis. When the last distance is 0, that is, when the axis of revolution passes through the centre of gravity, there will be no pressure on the axis arising from the centrifugal force, no matter what may be the mass of the body or its angular velocity. Such is the case of the earth revolving on its axis.

Principal Axes.

138. Suppose the axis about which a body revolves to become free, so that the body can move in any direction. If that axis be not one of symmetry, it will be pressed unequally in different directions by the centrifugal force, and will immediately alter its position. The body will for an instant rotate about some other line, which will immediately change its position, giving place to a new axis of rotation, which will instantly change its position, and so on, until an axis is reached which is pressed equally in all directions by the centrifugal forces of the elements. The body will then continue to revolve about this line, by virtue of its inertia, until the revolution is destroyed by the action of some extraneous force. Such an axis is called a principal axis of rotation. Every body has at least one such axis, and may have more. The axis of a cone or cylinder is a principal axis; any diameter of a sphere is a principal axis; in short, any axis of symmetry of a homogeneous solid is a principal axis. The shortest axis of an oblate spheroid is a principal axis; and it is found by observation that all of the planets of the solar system, which are oblate spheroids,