From these tables it will be seen that the resistance is less for tarred ropes, except those of very small size, than for dry ropes of equal radius.

The following rule for obtaining the resistance due to the rigidity of ropes is in general use: Multiply B in the preceding table by the weight in pounds, and add the product to A. Divide the sum by the effective radius of the sheave in inches, and the quotient will give the resistance in pounds.

Thus, if the weight to be raised is 500 pounds, and a 3-inch dry rope is used to lift it, passing round a sheave of five-and-a-half inches radius, the resistance due to rigidity is 29.8 pounds.

The effective radius of the sheave is its real radius plus the radius of the rope given in the table.

The following equation will determine the resistance where x= the weight required to be added to A' to move the machine. By the principle of the lever we have

(x+A') r=B' (r+b).

r expressing the radius of the pulley and b=EI. Hence

xr+Ar=B'r+B′b.

Since A'B'; A'r B'r. If these equals be taken from both, we have

xr=B'b..x=B22

r

Wherefore, if b is known, the corresponding resistance due to rigidity becomes apparent, since it is only required to consider the leverage of the resistance to be greater than it is by a certain quantity, and this quantity depends upon the curvature of the rope B ́E.

The curvature depends upon four things:-1st, the weight attached, let this w; 2nd, the radius of the rope-d; 3rd, the material of the rope-a; 4th, the radius of the wheel-r.

The determination of the above-mentioned quantity is performed by means of an empirical1 formula assumed to represent x, viz.

1 That is one, the truth of which is verified by experiment.

T

Here the letters m and n represent indeterminate numbers, the value of which, as also that of a, can only be determined by actual experiment. Let four sheaves be taken whose radii are r, ra, rb, rx, and let the rope whose rigidity is to be determined be laid over these in succession, and suspend weights equal to w, wa, wb, wc, and let the weights added to impart motion be x, x2, x1, x2. By substituting these in the above formula, we obtain—

From any three of these four equations the value of a, m, be ascertained. We have then

and n may

Thus we obtain the increase of leverage equivalent to the resistance due to the rigidity of a rope, its diameter and tension being known.

DIVISION II.

Kinetics.

CHAPTER I.

INTRODUCTORY.

Definition. Hitherto we have been considering forces in equilibrium, that is to say, that arrangement of forces which tends to keep a body in a state of rest. It now remains to consider the operation of forces not in equilibrium; or, in other words, those operations which impart motion to the body acted upon.

This portion of our subject is called Kinetics, which may therefore be defined as that branch of Dynamics which treats of the relation between forces and the motions they produce.

Divisions. This section of Dynamics will be considered under the following heads :

1. Falling Bodies.

2. The Pendulum.

5. Motion in a Circle.

6. Central Forces.

3. Motion down an Inclined 7. Impact.

Plane.

4. Projectiles.

8. Work.

Properties of matter.-Before we are in a position to explain the various forces producing motion, there are certain properties of matter which require to be understood. These are as follows:

The Quantity of Matter in a body is proportional to its volume and density conjointly. Thus: If A be a body having a volume V and a density D, and B a body hav

1 Gr., kineo, to move.

ing a volume equal to twice V and a density equal to the half of D, the quantities of matter in A and B are equal. Hence, when the products of the volume and density of bodies are equal, the quantities of matter in the bodies are also equal.

The Quantity of Motion is proportional to the quantity of matter and velocity. Thus: If A be a body having a quantity of matter Q and a velocity V, and B a body having a quantity of matter equal to twice Q and a velocity equal to half V, the quantities of motion in A and B are equal. The quantity of motion is called

Momentum.1

The Change of Momentum is proportional to the mass in motion and the change in the velocity, conjointly.

A ball of lead weighing 10 lbs. and moving with a velocity of 15 feet per second, would strike an obstacle with the same force as a ball weighing 50 lbs. moving at the rate of 3 feet per second. If we have but a small body with which to overcome a great resistance, but can impart to that body great velocity, so as to generate a great momentum, and then suddenly oppose the resistance by that momentum, we produce a force capable of overcoming the resistance. For example, in the pile-driving engine, the monkey 2 might lie on the top of the pile for an indefinite time without producing any effect. But by raising it to the top of the sheers and letting it fall upon the top of the pile, the momentum acquired by its motion and weight is such that the pile is driven into the earth.

Again, although we could never succeed in pushing a candle through a board, yet, when we impart great velocity to the candle, by firing it from a gun, we can drive it through an inch plank.

4

The sudden arresting of momentum is called percussion; and in our volume on Heat, it will be shown that the motion is changed by percussion into heat.

Kinetic Force.-In kinetics force is measured by a unit which, acting on a unit of mass, produces a unit of acceleration, or generates a unit of velocity in a unit of time.

The measure of kinetic force is, therefore, the quantity of motion produced in a unit of time; and the

1 L., moveo, to move, moving force. 2 The weight. scaffolding. L., percutio, to strike through.

4

• The

change in the velocity enables us to compare the magnitudes of different kinetic forces.

The amount of gravitation of various parts of the earth's surface was ascertained by comparing the velocities acquired in a unit of time by the falling of the same body in different localities.

Hence, if different masses, each acted upon by a force, acquire in equal units of time the same velocity, the forces are proportional to the masses, as is the case with the force of gravity on falling bodies.

CHAPTER II.

FALLING BODIES.

Illustrations.-If we throw a stone into the air, the force of gravity acting upon it will eventually cause it to descend to the earth. When bodies of different material fall through the air, they do not usually pass through the same number of units of space in the same time; and bodies of the same material fall through a greater or less number of units of space according to their shape.

Thus, a ball of lead and a piece of paper fall with different velocities. If an ounce of copper be beaten out into a sheet having a surface of one square foot, and another ounce be cast into a bullet, and the two bodies be let fall from the same height at the same instant, the ball will be found to reach the earth before the sheet of metal.

The difference is caused by the resistance opposed to the descent of the body by the upward pressure of the air on the surface opposed to it. This resistance varies with the form and dimensions of the body and with the velocity.

If, however, bodies which fall with different velocities through the air, such as a ball of lead and a feather,