radius AP, and b the height of the cylinder; then by the last proposition the moment of inertia of the cylinder CD, if it were solid, would beba,; also the moment of inertia of the cylinder PR, which is taken from this solid to form the hollow cylinder, would be ba1. Now let I represent the moment of inertia of the hollow cylinder CP, therefore I + {πbа„^={}πbα ̧*,

B

2

`. I=}πb(a‚1—a‚1)= {πb(a,‚2 — a‚2) (a‚2+a,2)=1πb(a,—a2) (a1+a2)(a22+a22).

Let the thickness a1-a, of the hollow cylinder be represented by c, and its mean radius (a,+a2) by R, therefore a1 =R+c, a2=R-1c.

Substituting these values in the preceding equation, we

*87. The moment of inertia of a cylinder about an axis passing through its centre of gravity, and perpendicular to its axis of symmetry.

Let AB be such an axis, and let PQ represent a lamina

P

M

C

R

A

B

contained between planes perpendicular to this axis, and exceedingly near to each other. Let CD, the axis of the cylinder, be represented by b, its radius by a, and let CM=x. Take Ax to represent the thickness of the lamina, and let MP=y. Now this lamina. may be considered a rectangular parallelopiped traversed through its centre of gravity by the axis AB; therefore by equation (62) its moment of inertia about that axis is represented by (Ax) b (2y) {b2 + (2y)2} = b {b2y + 4y3} Ax. Now the whole moment of inertia I of the cylinder about AB is evidently equal to the sum of the moments of inertia of all

such laminæ ;

.•. I= {bΣ{b2y + 4y3} Ax= {bƒ{b2y+4y3)dx.

G 4

Also, since x and y are the co-ordinates of a point in a circle from its centre, therefore y=(a2-x2). Substituting this value of y, and integrating according to the well known rules of the integral calculus, we have

*88. The moment of inertia of a cone about its axis of

symmetry.

The cone may be supposed to be made up of laminæ, such as PQ, contained by planes perpendicular to the axis of symmetry AB, and each having its centre of gravity in that axis. Let BP=x, and let Ax represent the thickness of the lamina, and y its radius PR. Then, since it may be considered a cylinder of very small height,

B

R

its moment of inertia about AB (equation 64) is represented by yAx. Now the moment of inertia I of the whole cone is equal to the sum of the moments of all such elements,

Let the radius of the base of the cone be represented by

ь

a, and its height by b; therefore-= therefore Ax=-▲y;

y a
b

a

(67).

89. The moment of inertia of a sphere about one of its

diameters.

Let C be the centre of the sphere and AB the diameter

about which its moment is to be determined. Let PQ be any lamina contained by planes perpendicular to AB; let CM=x, and let Ax represent the thickness of the lamina, and y its radius; also let CA=a; then since this lamina, being exceedingly thin, may be considered a cylinder, its moment of inertia about the axis AB is (equation 64) у'Ax; and the moment of inertia I of the whole sphere is the sum of the moments of all such laminæ,

Now by the equation to the circle y2=a2x2, therefore y1=a1 — 2a2x2+x. If this value be substituted for y1, and the integration be completed according to the common methods, we shall obtain the equation,

90. The moment of inertia of a cone about an axis passing through its centre of gravity and perpendicular to its axis of symmetry.

A

G

R

of

Let CD be an axis passing through the centre of gravity G of the cone, and perpendicular to its axis symmetry, and let GP the distance of the lamina from G, measured along the axis, be represented by x; also let the thickness of the lamina be represented by Ar. Now this lamina may be considered a cylinder of exceedingly small thickness. If its radius be represented by y, its moment of inertia about an axis parallel to CD passing through its centre, is therefore (equation 66) represented by \xy2 {y2+ }(Ax)2} Ax, or if Ax be assumed exceedingly small, it is represented by yAx. Now this being the moment of the lamina about an axis parallel to CD, passing through its centre of gravity, and the distance of this axis from CD being, and also the volume of the lamina being xy'▲x, it

follows (equation 58), that the moment of the lamina about CD is represented by y2x2Ax+4xу1▲x = {y2x2 + 4y1} ▲x. Now the moment I of the whole cone about CD equals the sum of the moments of all such elements,

.•. I=#Σ(y2x2 + 4y1)Ax.

Now if a be the radius of the base of the cone and bits height, then since BG=3b,

91. The moment of inertia of a segment of a sphere about a diameter parallel to the plane of section.

Let ADBE represent any such portion of a sphere, and AB a diameter parallel to the plane of section.

B

Let CD=a, CE=b, and let PQ be any lamina contained by planes parallel to the plane of section let the distance of the lamina from C=x, and let its thickness be Ar and its radius y. Then considering it a cylinder of exceeding small thickness, its moment of inertia about an axis passing through its centre of gravity and parallel to AB, is represented (equation 66) by πу2 {y2+}(Ax)2} Ax, or (neglecting powers of Ar above the first) by y1Ax. Hence, therefore, the moment of this lamina about the axis AB is represented (equation 58) by яу(Ax)x2 + уAx, or by π (y2x2 + 4y1} Ax ; now the whole moment I of inertia of ADBE about AB is evidently equal to the sum of the moments of all such laminæ,

.• . I=n& {y2x2 + {y1}▲x=«S (y2x2 + \y*}dæ.

Now y2=a2x2, therefore y2x2 + ¦ÿ1= 4 {2a2x2—3x1+a1} . Substituting this value in the integral and integrating, we have

I=zzπ {16a2+15a1b+10a2b3—9b5}

.

(70).

THE ACCELERATION OF MOTION BY GIVEN MOVING FORCES.

92. IF the forces applied to a moving body in the direction of its motion exceed those applied to it in the opposite direction (both sets of forces being resolved in the direction of a tangent to its path), the motion of the body will be accelerated; if they fall short of those applied in the opposite direction, the motion will be retarded. In either case the excess of the one set of forces above the other is called the MOVING FORCE upon the body: it is measured by that single pressure which being applied to the body in a direction opposite to the greater force, would just balance it; or which, had it been applied to the body (together with the other forces impressed upon it) when in a state of rest, would have maintained it in that state; and which therefore, if applied when its motion had commenced, would have caused it to pass from a state of variable to one of uniform motion. Thus the moving force upon a body which descends freely by gravity, is measured by its weight, that is, by the single force which, being applied to the body before its motion had commenced in a direction opposite to gravity, would just have supported it, and which being applied to it at any instant of its descent, would have caused its motion at that instant to pass from a state of variable to a state of uniform motion. If the resistance of the air upon its descent be taken into account, then the moving force upon the body at any instant is measured by that single pressure which, being applied upwards, would, together with the resistance of the air at that instant, just balance the weight of the body.