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In another form of the compensating pendulum, the ball is supported by a framework composed of rods of different metals, so adjusted that the downward expansion of one part is exactly compensated by the upward expansion of the other part.

In the form shown in Fig. 43, called the gridiron pendulum, there are five steel bars expanding downward and four brass bars expanding upward. As the relative expansibility of brass compared with steel is as 100 to 61, the length of the steel bars is 100 that of the brass.

75. Length of the Seconds Pendulum. The length of the pendulum vibrating seconds has been very accurately determined. At the same place it is invariable, but it varies with the latitude. At the equator it is 39.0217 inches; at New York, 39.10237 inches; at Spitzbergen, 39.21614 inches. The cause of this variation is the difference in the force of gravity in different places, due to the spheroidal shape of the earth.

[graphic]

Fig. 43.

The polar diameter of the earth being twentysix miles shorter than the equatorial diameter, any point on the surface of the earth near either pole is nearer the centre, and the force of terrestrial gravity is stronger than at points on or near the equator. Consequently, a pendulum which vibrates seconds at the equator, on being carried to a latitude of 40° to 50°, is more strongly acted upon by gravity, and vibrates more rapidly. In order, therefore, that it may continue to make exactly one vibration in each second, the rapidity of vibration must be diminished by increasing the length of the pendulum.

Summary.

The Pendulum.

Vibration or Oscillation.

Illustration.

Simple Pendulum.

Compound Pendulum.

[blocks in formation]

76. Work.

The term work as used in mechanics means

the production of motion against resistance.

It is obvious that this definition will apply not only to the labor of men and animals, but to the action of forces of other kinds as those of wind, water, and steam when employed in overcoming resistance.

In this sense, drawing loads, raising weights, pumping water, forging iron, pressing cotton, etc., are all examples of work, whatever may be the forces employed in the various operations.

77. Measurement of Work. The work done in raising a weight to a given height is generally taken as a standard for the measurement of work.

In this country and in England the unit of work commonly adopted is the foot-pound.

This is the amount of work required to raise one pound one foot against the force of gravity.

The unit of the Metric System is the work required to

raise one kilogram to a height of one meter. It is called a kilogram-meter.

To find a numerical expression for the work in a given example, we inultiply the number of weight units raised by the number of linear units in the vertical height to which the body is raised. A weight of 20 lbs. raised 4 feet, or a weight of 4 lbs. raised 20 feet represents 80 foot-pounds. A weight of 25 kilograms raised 5 meters represents 125 kilogram-meters.

78. Horse-Power.

It has been estimated that the strength of a horse is on the average, sufficient to raise 33,000 pounds vertically through one foot in a minute; hence a horse-power is a power which can perform 33,000 units of work in a minute.

The capacity of steam-engines and other powerful machines is generally rated by horse-powers; thus, an engine is said to be of ten horse-power if it is capable of doing work equivalent to raising 33,000 lbs. 10 feet in one minute, or 330,000 lbs. one foot in a minute.

The time required for the work is an essential part of the calculation. If an engine can do 33,000 units of work in half a minute, it is of two horse-power; if it can do the same work in one second, it is of sixty horse-power.

79. Energy is the power of doing work, that is, of overcoming resistance. Any moving body can overcome resistance, and therefore possesses a certain amount of energy. The amount of energy in a moving body depends upon its weight and velocity. The direction in which it moves makes no difference in the energy with which it acts. If its energy is expended in lifting itself against the force of gravity, we can, if its weight and velocity are known, determine the amount of this energy in foot-pounds, or kilogram-meters.

To do this we have simply to find the vertical height to which the given velocity would lift the body, and multiply the weight by the height. Let m = the mass of a body, and v the velocity with which it is moving, and its energy will be expressed by the formula m2; that is, its energy is equal to one half its mass multiplied by the square of its velocity.

80. Kinetic and Potential Energies. To understand these two types of energy, let us consider the case of a heavy body thrown directly upward into the air. As it begins to rise, it has a certain amount of energy depending upon the velocity with which it moves. This is its energy of motion. As it continues to rise, its velocity, and consequently its energy of motion, decreases, until at the highest point which it reaches it has no longer any energy of motion. But in consequence of its elevated position, it has the power of doing work in its fall to the earth again; that is, it has energy of position.

Energy of motion is called kinetic energy.

Energy of position is called potential energy.

In the case just given, the sum of the two types of energy remains the same for every position of the body; for, as it rises, kinetic energy decreases, and potential energy increases in exactly the same proportion, while in its descent potential energy decreases and kinetic energy increases till the body comes to rest in its original position.

A body may have energy of position from other causes than being raised to a height.

A bow that is bent, the mainspring of a watch that is wound up, or any body in which reserved force is stored up has potential energy.

Summary.

Work.

Definition of Work.
Examples.

Measurement of Work.
Unit of Work.

The Foot-Pound.

The Kilogram-Meter.

Horse-Power.

Energy.

Measurement of Energy.

Kinetic Energy.

Potential Energy.

Illustration.

Examples of Potential Energy.

CHAPTER III.

APPLICATION OF PHYSICAL PRINCIPLES TO MACHINES.

SECTION I.

GENERAL PRINCIPLES.

B1. A Machine is a contrivance by means of which a force applied at one point is made to produce an effect at some other point.

The force applied is called the power, and the force to be overcome is called the weight, or load.

82. Motors.

- The working of a machine requires a continued application of power. The source of this power is

called the MOTOR.

Some of the most important motors are muscular effort, as exerted by man or beast, in various kinds of work; the weight and impulse of water, as in water-mills; the impulse of air, as in wind-mills; the elastic force of springs, as in watches; the expansive force of vapors and gases, as in steam and hot-air engines. The last is, perhaps, the most useful of the motors mentioned.

83. Object and Utility of Machines. - The object of a machine is to transmit the power furnished by the motor, and to modify its action in such a manner as to cause it to produce a useful effect.

In no case does a machine add anything to the power applied to it; on the contrary, it absorbs more or less of this power, according to the nature of the work to be done and the connection existing between the parts.

Some of the circumstances which cause an absorption of power

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