Ex. 736.-What is the length of a simple pendulum which at Greenwich oscillates in 1 seconds? How much shorter is the simple pendulum which at Rawak (Table XV.) oscillates in the same time?

Ex. 737.—A pendulum whose length is L makes m oscillations in one day; its length changes, and it is now observed to make m+n oscillations in one day; show that its length has been diminished by a part equal to 2n L (nearly).

m

[Since a mean solar day contains 86,400 seconds, we have

Ex. 738.-A pendulum in a certain place makes in one day m oscillations; on transporting it to another place it is found to have the same length but to lose n oscillations a day; show that the force of gravity has been dimi

Ex. 739.-Given the lengths of the seconds pendulums at Greenwich and Paris respectively (see Table XV.), find how many oscillations a day the Greenwich pendulum would make at Paris. Ans. 86,387.

Ex. 740.—Given that a pendulum oscillating seconds at the mouth of a coal pit gains 2.24 seconds per diem when removed to the bottom of the shaft; determine the decrease of the force of gravity. Ans. 0.0016.

Ex. 741.-A body leaves c (fig. 166) with such a velocity that A is the highest point it reaches, the arc AC being small. Let the arc ac be denoted by 8, and the time of a double oscillation by T. Show that at a time t its distance s from c is given by the formula

[From Prop. 29 it is plain that the time of describing CP bears to that of describing ca the same ratio that the arc cp bears to cpD or that the angle cop bears to π.

Ex. 742.-If v is the velocity at c, and v that at a time t, show that

Ex. 743.-In Ex. 741 and 742 obtain the value of s and v, when t is reckoned from the instant the body is at A.

[For t write t+T.]

Ex. 744.-A body vibrates in a small circular arc, the velocity at the lowest point being v; when at its lowest point it has communicated to it a velocity v at right angles to its plane of vibration; show that its plane of vibration is turned through an angle of 45°, its arc of vibration increased from s1 to s1√2, and its time of oscillation sensibly unchanged.

Ex. 745.-In the last case, if the velocity is communicated to the body when at A, show that it will now describe a horizontal circle round c, whose radius is s1 and time of description 2π

Ex. 746.-If the angle asc (fig. 166) is denoted by 0, the mass of the body by м, and the tension of the thread when the body is at c by T, show that T-Mg(3-2 cos 0).

=

Ex. 747.—In Ex. 741 show that the acceleration along the curve is 98.

Hence when a body, whose mass is м, vibrates under the action of a force Hs, where s is the distance of м from the middle point of the arc or line of vibration, show that the time of vibration equals M

Π

133. Longitudinal Vibrations of a Rod.—If there is a rod whose length is L, area of section K, and modulus of elasticity E, and if to the end of it is attached a weight Q (which we will suppose to be so large that the weight of the rod can be neglected), then if the rod is allowed to lengthen slowly, Q will descend through a small space l equal to and will continue at rest (see Art. 6 and

LQ

KE

Ex. 149); if, however, it is allowed to descend at once, a certain number of units of work will be accumulated in it when Q has descended through the space l, so that it will continue to descend till the resistance to further elongation shall have destroyed them, and then a contraction will ensue, and thus Q will vibrate in a vertical

line about the point (A), at which in the former case it would have come to rest.

Ex. 748.-Show that when the weight, which contains o lbs. of matter, is at a distance s from a it is moving under a force that varies as s, and that the time in which it proceeds from the highest to the lowest point is

Ex. 749.-In the last Example suppose a to be at a distance s below a, determine the number of units of work accumulated in it at that instant, and show that its velocity (v) is given by the equation

[See Ex. 149. The student must bear in mind that g stands for 32.1912, the quantity gKE being, in fact, the Modulus of Elasticity reckoned with reference to the whole cross-section of the rod and in absolute units. The same is true of the answer to Ex. 748. In Ex. 750 g stands for the accelerative effect of gravity, whatever it may be.]

Ex. 750.-If a cylinder whose height is h and specific gravity s floats with its axis vertical in a fluid whose specific gravity is s1, show that if it is depressed through any distance the time in which it will rise from its point of greatest depression to its greatest height is constant, and will be given by the formula

CHAPTER V.

THE MOMENT OF INERTIA.

Def.-If we conceive a body to consist of a large number of heavy points, and multiply the mass of each by the square of its perpendicular distance from a given line or axis, the sum of all these products is the moment of inertia of the body with respect to that axis.

134. Properties of the Moment of Inertia. It will appear hereafter that the moment of inertia is a quantity that enters nearly every question in which the rotatory motion of a body is concerned; the present chapter will be devoted to proving some of its properties, and ascertaining its magnitude in certain particular cases. The first property we shall notice is one that follows immediately from the definition. Since the mass of a particle and the square of its perpendicular distance from a given axis are essentially positive, their product must be so too; consequently if we conceive any group of heavy points to consist of two or more subordinate groups, the sum of the moments of inertia of these separate groups with respect to a given axis will equal that of the whole group with respect to the same axis: hence if a body can be divided into a certain number of parts, and their respective moments of inertia are known with respect to a certain axis, that of the whole body, with respect to that axis, is found by adding them together.

Proposition 30.

FIG. 167.

If I is the moment of inertia of any body whose mass is м, about an axis passing through its centre of gravity, and I, the moment of inertia of the same body about a parallel axis situated at a perpendicular distance h from the former, then

I1 =I+Mh2.

N

Suppose the axes to be perpendicular to the plane of the paper, let the axis which passes through the centre of gravity meet that plane in o, and let the other meet it in o1; let p be one of the points of which the body is conceived to be made up, and let its mass be m1; join PO, PO1, and 0 01, and draw PN at right angles to o o1; then (Eucl. II. 13)

0

01p2=0p2+0012 — 2 0 01. o N.

Let op="1, 01Pr1, ON=x1, and oo1=h, then

m1 r{'2=m, r12+m ̧ h2 −2m, hx,

(1)

and the same algebraical formula will be true whatever be the position of P; hence if m2, r2', 12, X2, M3, r3′, 139 X39 &c., are the magnitudes corresponding to other points,

...

and by the properties of the centre of gravity (Prop. 16)

since the weight of each particle is proportional to its mass.