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feet, and the radius of whose base is 4 feet, the weight of the material being estimated at 100 lbs. per cubic foot?

SOLUTION.

The weight of the cone is equal to

TX 42 × 4 x 100 lbs. 20106.24 lbs.

If the cone turns about a tangent to its base, since the centre of gravity is 3 feet from the base, it will be,

√3+425 feet from the tangent.

The centre of gravity, at its highest point, will, therefore, be 5 feet from the horizontal plane. It must then be raised 2 feet. Hence, the required quantity of work is equal to

20106.24 lbs. X 2 ft. 40212.48 lbs. ft. Ans.

14. To show that the work required for overturning similar solids, similarly placed, varies as the fourth powers. of their homologous lines.

SOLUTION.

Denote the altitudes of the centres of gravity, by y and ry, the distances from the directions of the weights to the lines about which they turn, by x and rx, and their weights, by w and r3w.

The quantity of work required to overturn the first, will be,

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The quantity of work required to overturn the second,

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Rotation.

146. When a body restrained by a fixed axis, about which it is free to turn, is acted upon by a force, it will, in general, take up a motion of rotation, or revolution. In this kind of motion, each point of the body describes a circle, whose centre is in the axis, and whose plane is perpendicular to the axis. The time of a complete revolution being the same for each particle, it follows, that the velocities of the different particles will be proportional to their distances from the axis. The velocity of any particle will be equal to its distance from the axis multiplied by the angular velocity (Art. 122).

Quantity of work of a Force producing Rotation.

147. If a force is applied obliquely to the axis of rotation, we may conceive it to be resolved into two components, one parallel, and the other perpendicular to the axis of rotation. The effect of the former will be counteracted by the resistance offered by the fixed axis; the effect of the latter in producing rotation will be exactly the same as that of the applied force. We need, therefore, only consider those components whose directions are perpendicular to the axis. of rotation.

E

B

P

Let P represent any force whose line of direction is perpendicular to the axis, but does not intersect it. Let O be the point in which a plane through P, perpendicular to the axis, intersects it. Let A and C be any two points whatever, on the line of direction of P. Suppose the force P to turn the system through an infinitely small angle, and let B and D be the new positions of A and C. Draw OE, Ba, and De respectively perpendicular to PE; draw also, AO, BO, CO, and DO. Denote the distances OA, by r, OC, by r', OE, by p, and the path described by

Fig. 128.

a point at a unit's distance from O, by '. Since the angles

AOB, and COD are equal, from
the nature of the motion of rota-
tion, we shall have, AB = r',
and CD = r''; and since the
angular motion is infinitely small,
these lines may be regarded as
straight lines, perpendicular re-
spectively to OA and OC.
ABa and CDc, we have,

E

Fig. 128.

B

Р

A

From the right-angled triangles

Aare'cos BAa, and Сс = r'e'cosD Cc.

In the right-angled triangles ABa, and OAE, we have AB perpendicular to OA, and Aa perpendicular to OE; hence, the angles BAa, and AOE, are equal, as are also their cosines; hence, we have,

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Substituting in the equations just deduced, we have,

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The first member of the equation is this quantity of work of P, when its point of application is at A; the second is the quantity of work of P, when its point of application is at C. Hence, we conclude, that the elementary quantity of work of a force applied to produce rotation, is always the

same, wherever its point of application may be taken, provided its line of direction remains unchanged.

We conclude, also, that the elementary quantity of work is equal to the intensity of the force multiplied by its lever arm into the elementary space described by a point at a unit's distance from the axis.

If we suppose the force to act for a unit of time, the intensity and lever arm remaining the same, and denote the angular velocity, by e, we shall have,

Q' = Pps.

For any number of forces similarly applied, we shall have,

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If the forces are in equilibrium, we shall have (Art. 48), (PP) = 0; consequently, Q = 0.

Hence, if any number of forces tending to produce rotation about a fixed axis, are in equilibrium, the entire quantity of work of the system of forces will be equal to 0.

148.

Accumulation of Work.

When a body is put in motion by the action of a force, its inertia has to be overcome, and, in order to bring the body back again to a state of rest, a quantity of work has to be given out just equal to that required to put it in motion. This results from the nature of inertia. A body in motion may, therefore, be regarded as the representation of a quantity of work which can be reproduced upon any resistance opposed to its motion. Whilst the body is in motion, the work is said to be accumulated. In any given. instance, the accumulated work depends, first, upon the mass in motion; and, secondly, upon the velocity with which it moves.

Take the case of a body projected vertically upwards in vacuum. The projecting force expends upon the body a quantity of work sufficient to raise it through a height equal

to that due to the velocity of projection. Denoting the weight of the body, by w, the height to which it rises, by h, and the accumulated work, by Q, we shall have,

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Denoting the mass of the body by m, we shall have,

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m = (Art. 11), and, by substitution, we have, finally,

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If the body descends by its own weight, it will have impressed upon it by the force of gravity, during the descent, exactly the same quantity of work as it gave out in ascending.

The amount of work accumulated in a body is evidently the same, whatever may have been the circumstances under which the velocity has been acquired; and also, the amount of work which it is capable of giving out in overcoming any resistance is the same, whatever may be the nature of that resistance. Hence, the measure of the accumulated work of a moving mass is one-half of the mass into the square of the velocity.

The expression mv2, is called the living force of the body. Hence, the living force of a body is equal to its mass, multiplied by the square of its velocity. The living force of a body is the measure of twice the quantity of work expended in producing the velocity, or, it is the measure of twice the quantity of work which the body is capable of giving out.

When the forces exerted tend to increase the velocity,

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