One Erg. An erg is a very small amount of work, hence a multiple of it is taken as the usual practical unit of work, viz. 107 ergs or 10 million ergs, and this is called one Joule. 3,600,000 joules are called one Board of Trade unit of work or one B.T.U. The Board of Trade unit of work is, therefore, equal to 36 billion ergs.

When any agent is working against a force, the most important question is generally to determine the rate at which it works. This rate of working is called its Activity or Power.

The C.G.S. unit of power is an activity of one erg done per second. The work done in lifting a mass of one gramme one centimetre against gravity, is 981 ergs. The practical unit of activity or power is equal to 107 or 10 million ergs done per second, and this is called a Power of one Watt.

One kilowatt is 1000 watts, and a kilowatt is nearly equal to 13 horse-power.

One horse-power is equal to 33,000 foot-lbs. per minute, or to the power required to lift 33,000 lbs. I foot high in 1 minute against the force of gravity.

If a body is at rest or moving uniformly, and we act upon it by any force so as to change its velocity, we are said to do work against the force of inertia, and the work so done is measured by the increase produced in the value of the kinetic energy of the body. This kinetic energy is measured at any instant by the product of the mass of the body and half the square of its velocity.

The usual British unit of work is the foot-pound, and it is defined as the work done in lifting one lb. one foot high against the force of gravity. As, however, the force of gravity is different at different places on the earth's surface, being greater by about one-half per cent. at the poles than at the equator, the foot-pound is not an absolutely defined unit unless some place is named at which the pound is lifted.

We have, however, the following practical equivalents:

I joule
I watt

I kilowatt

= 10' ergs = 7373 foot-lb.

= 1 joule per second = 10' ergs per second. 13 horse-power nearly.

746 watts = 1 horse-power = 550 foot-lbs. per second.

The energy or work required to heat 1 gramme of water I degree C. in temperature in the neighbourhood of 10° C. has been determined to be 42 million ergs, or 42 joules or 3 096 foot-lbs. This amount of heat is called 1 gramme-centigrade degree unit of heat, and the value 4 2 joules is called the Mechanical Equivalent of Heat. An amount or quantity of energy of any kind is measured in ergs or joules.

The student will notice that in this case most of these derived units, viz. in the case of velocity, acceleration, force and activity, the measure of them is a rate at which some other quantity is changed. Velocity is the time-rate of change of position. Acceleration is the time-rate of change of velocity. Force is the time-rate of change of momentum. Activity or power is the timerate of change of work or energy.

There are some other terms which it will be convenient to define at this point. When any body is acted upon by two equal and parallel forces not acting in the same straight line, these constitute what is known as a couple or torque, and the mechanical value of the couple or torque in producing twist or rotation is measured by the product of the value of either force and the vertical distance between the forces. The effect of a couple acting on a body is to cause rotation or twist round an axis. The unit couple is that due to two forces each of one dyne acting at a distance of one centimetre and causing twist round an axis perpendicular to their own direction.

In many problems in physics we have to deal with. the case of a body rotating or swinging round an axis,

such as a fly-wheel rotating or a pendulum swinging. An important quantity which then presents itself is that of the Moment of Inertia of the body. As above stated, this quantity is calculated by assuming the body to be divided into equal and very smal! parts, taking the product of the mass of each part and the square of its vertical distance from the axis of rotation, and adding together all these products. The moment of inertia and the angular velocity of a rotating body enter into such rotational problems just as the mass and linear velocity do into the corresponding problems in the movement of a particle of matter. Thus the product of the moment of inertia and the angular velocity is called the Angular Momentum. The product of the moment of inertia and half the square of the angular velocity is called the Angular or Rotational Energy, and the rate of change of angular momentum is the measure of the torque or couple causing rotation. The dynamical measure of the torque or couple is the rate at which it changes the angular momentum of the system estimated round the axis of the torque.

The moment of a force with respect to any axis perpendicular to its direction is measured by the product of the numerical value of force and the vertical distance between the axis and the line representing the force.

In any system of bodies rotating round a centre, such as the planets and the sun round their common centre of mass, no interaction between the different bodies can change the total angular momentum of the system. This last can only be altered by some torque acting on the system from without. This fact is one of fundamental importance. It is called the principle of the Conservation of Angular Momentum, or of the Moment of Momentum. It is in reality only the fundamental fact concerning matter, viz. that it cannot change its own state of rest or motion, applied to the particular case of rotation of a number of masses.

CHAPTER III.

MAGNETIC FORCE AND MAGNETIC FLUX.

§1. A Unit Magnetic Pole.-Let a long thin steel wire or knitting needle be uniformly magnetised, and then broken in the middle. At the point of rupture it will, as already described, develope new, equally strong north and south magnetic poles, which would, if held near to each other, mutually attract.

Imagine these poles placed one centimetre apart, and that the pull or attractive force between the poles is measured by a very delicate spring balance. Furthermore, suppose that this force proves to be just one dyne or a dynamical unit of force, as defined in the previous chapter. Then these two equal magnetic poles, which attract each other with a force of one dyne when placed one centimetre apart, are called Unit Magnetic Poles. The strength of a magnetic pole is numerically measured by the number of unit magnetic poles to which that particular pole considered is magnetically equivalent. Thus any given magnet may have a pole of strength equal, say, to 10 units; and this would denote that in all respects the given pole was equivalent to ten unit magnetic poles placed together; assuming that these did not in any way affect each other in power by being put together.

§ 2. Moment of a Magnet.-By the phrase moment of a magnet is meant the value of the product of the strength of either pole of the magnet and the shortest distance between the poles. Thus, consider a strongly magnetised steel wire. Let it be 10 centimetres long, and let it have pole strength represented by 10; that is,