(4) One end A of a rod AB is attached to a hinge A in a vertical axis, and the other end B is connected with a weight P by means of a fine string passing through a small hole in the axis at a distance, above A, equal to AB: supposing the rod to revolve about the axis, to determine its angular velocity in order that it may be inclined during the whole motion at an

π

angle to the vertical line drawn downwards from A.

4

Let 2a denote the length of the rod, and W its weight: then, w representing the angular velocity,

P
IP √(4.

(4 − √/8)} .

(5) A rod revolves freely about a fixed point at its upper end so as to be always inclined to the vertical at a given angle: to find its angular velocity about the vertical through its higher end, and the direction of the pressure on the fixed point.

Let 2a be the length of the rod and a the given angle: then the angular velocity will be equal to

and, being the inclination of the direction of the pressure to the vertical,

tany

3
tan a.
4

Griffin: Solutions of the Examples on the motion

of a rigid body, p. 35.

(6) An elastic string, the weight of a unit of length of which in its natural state is w, is placed within a circular tube of radius a, which revolves uniformly about a vertical diameter:

the modulus of elasticity is wa and the angular velocity (2)*:

if the string occupy the upper half of the tube, to find the natural length of the string.

The natural length of the string is equal to the length of the diameter of the tube.

(7) A square lamina revolves about an edge, which is fixed in a vertical position to ascertain the angular velocity in order that the resultant pressure on the axis of revolution may pass through the highest point.

If 2a be the length of an edge, the required angular velocity is equal to

(2) * .

(8) A carriage moves on a railroad with a given velocity round a curve of given radius: to find the amount by which the outer rail must be elevated above the inner one in order that the carriage may not be overturned towards the outside.

We will suppose the radii of the circles described by the molecules of the carriage to be the same, as will be approximately the case in railroads.

=

Let 26 the breadth of the road between the rails, a the distance of the centre of gravity of the carriage from the road, r = the radius of the curve, v = the velocity of each molecule of the carriage, and the inclination of the road to the horizon: then

=

(9) A thin book lies on one of the faces of a desk to find the greatest angular velocity round a vertical axis which can be given to the desk without throwing off the book.

=

Let a the inclination of the desk to the horizon, a the length of the book; and, the book being supposed to be placed symmetrically on one face of the desk, let c = the distance of its lower edge from the axis of revolution, the required angular w =

velocity. Then, the book being supposed to be moveable about its lower edge, which is kept at rest by the ledge of the desk, 3g cot a

=

3c- 2a cos a

(10) A circular disc is capable of motion about a horizontal tangent, which rotates with a uniform angular velocity about a vertical axis through the point of junction, which is fixed: to find the angular velocity of the tangent in order that the inclination of the disc to the horizon may have a given constant value.

Let a be the radius of the disc and a the inclination of the disc to the horizon: then the required angular velocity is equal

to

(sa hima).

5a sin

(11) A Ring, surrounding a Planet, revolves uniformly about a diameter passing through the common centre of the Ring and the Planet to determine the form of the Ring in order that the tangential stress may be the same at all points.

If w= the angular velocity, μ=the attraction of the planet at a unit of distance, 2a = the diameter of revolution; then, the prime radius vector being supposed to be coincident with the diameter of revolution, the equation to the ring will be

1

1

α

r

=

r2 sin2 0. 2μ

SECT. 3. Centre of Oscillation.

Conceive a body of any figure, acted on by gravity, to be oscillating about a fixed horizontal axis AB (fig. 188); let G be the centre of gravity of the body; draw GO at right angles to AB. Produce OG to a point C such that

where h = OG and k = the radius of gyration of the body about an axis through G parallel to AB; then, if the whole mass of the body be collected at the point C, the period of its oscillations about AB will be the same as before. The point C is called the Centre of Oscillation or of Agitation.

The theory of the Centre of Oscillation of bodies originated in questions addressed, about the year 1646, by Mersenne to the mathematicians of his day, who were called upon by him to exert their ingenuity to discover the time of oscillation of bodies moveable about horizontal axes. It is rather singular that all those who first attempted the solution of this celebrated problem, among whom Mersenne' himself is to be numbered, together with Descartes, Roberval3, Wallis*, and Fabri3, tacitly supposed the Centre of Oscillation to be coincident with the Centre of Percussion; a supposition which, although true, is by no means obvious without a rigorous demonstration. On the strength of this assumption, however, the Centre of Oscillation was correctly determined in the case of certain figures. Descartes gave a true solution of the case where a plane area oscillates in planum, but failed in the case of solid bodies and of plane areas oscillating in latus. Roberval assigned correctly the position of the Centre of Oscillation, not only of plane areas oscillating in planum, but also in certain instances of oscillation in latus, while together with Descartes he failed to give a correct solution of the problem in the case of solid figures. The labours of Huyghens, who in his earlier efforts to obtain a solution of Mersenne's problem had been utterly baffled, were at length crowned with success, and accordingly in the fourth part of his Horologium Oscillatorium, which appeared in the year 1673, was given the first rigorous and general investigation of the Centre of Oscillation. The two following axioms constitute the basis of his researches: first, that the centre of gravity of a system

1 Mersenni Reflexiones Physico-Mathematica, Cap. XI. et XII. 2 Lettres de Descartes, Tom. III. p. 487, &c.

3 Lettres de Descartes, ib.

4 Mechanica, sive De Motu.

5 Tract. de Motu, Append. Physico-Math. De Centro Percussionis.

of heavy bodies cannot of itself rise to an altitude greater than that from which it has fallen, whatever change be made in the mutual disposition of the bodies; and, secondly, that a compound pendulum will always ascend to the same height as that from which it has descended freely. Some years after the publication of the Horologium Oscillatorium, the truth of these fundamental axioms, which although true, it must be admitted, are not sufficiently elementary, was called in question by the Abbé Catelan', who substituted certain frail theories of his own in place of the valuable researches of Huyghens. The attention of the mathematicians of the day having been more closely directed to the subject by the controversy which arose between Huyghens and Catelan, the views of Huyghens received ample corroboration from the more elementary investigations of L'Hôpital, James Bernoulli, and other mathematicians. For information respecting the subsequent history of Mersenne's problem, the reader is referred to the Chapter on D'Alembert's Principle.

(1) To find at what point of the rod of a perfect pendulum must be fixed a given weight of indefinitely small volume, so as to have the greatest effect in accelerating the pendulum.

Let m be the mass of the bob of the perfect pendulum, and a its length; m' the mass of the given weight, and a' the distance of its point of attachment from the centre of suspension; 7 the distance between the centre of suspension and the centre of oscillation of the complex pendulum. Then we shall have, m and m' being both of indefinitely small volume,

ma2 + m'a22 1=

ma + m'a'

Now the shorter the rod of a perfect pendulum, the shorter will be the time of its oscillations: hence I must be a minimum: differentiating then with respect to a' we get