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CHAPTER III.

MECHANICAL UNITS.

Value of g.

23. ACCELERATION is defined as the rate of increase of velocity per unit of time. The C.G.S. unit of acceleration is the acceleration of a body whose velocity increases in every second by the C.G.S. unit of velocity-namely, by a centimetre per second. The apparent acceleration of a body falling freely under the action of gravity in vacuo is denoted by g. The value of g in C.G.S. units at any part of the earth's surface is approximately given by the following formula,

g=980 6056 -2.5028 cos 2λ - ·000003h,

A denoting the latitude, and h the height of the station (in centimetres) above sea-level.

The constants in this formula have been deduced from numerous pendulum experiments in different localities, the length of the seconds' pendulum being connected with the value of g by the formula g = π2l.

Dividing the above equation by 2 we have, for the length of the seconds' pendulum, in centimetres,

7=99.3562·2536 cos 2λ - 0000003.

At sea-level these formulæ give the following values for

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The difference between the greatest and least values

(in the case of both g and 7) is about of the mean

value.

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24. The Standards Department of the Board of Trade, being concerned only with relative determinations, has adopted the formula

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A denoting the latitude, h the height above sea-level, R the earth's radius, g。 the value of g in latitude 45° at sealevel, which may be treated as an unknown constant multiplier. Putting for R its value in centimetres, the formula gives

g=g。(100257 cos 2λ - 1.96h × 10−o), where h denotes the height in centimetres.

The formula which we employed in the preceding section gives

2h

g=g.(1 – 00255 cos 2A) (1 – 14).

R

As regards the factor dependent on height, theory indi

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as its correct value for such a case as that of

a balloon in mid-air over a low-lying country; the value

1

5 h

4 R

may be accepted as more correct for an elevated

plateau on the earth's surface.

Force.

25. The C.G.S. unit of force is called the dyne. It is the force which, acting upon a gramme for a second, generates a velocity of a centimetre per second.

It may otherwise be defined as the force which, acting upon a gramme, produces the C.G.S. unit of acceleration, or as the force which, acting upon any mass for 1 second, produces the C.G.S. unit of momentum.

To show the equivalence of these three definitions, let m denote mass in grammes, v velocity in centimetres per second, t time in seconds, F force in dynes.

Then, by the second law of motion, we have

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that is, if a denote acceleration in C.G.S. units, a=

hence, when a and m are each unity, F will be unity.

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Again, by the nature of uniform acceleration, we have v = at, v denoting the velocity due to the acceleration a, continuing for time t.

Hence we have F: = ma =

mv t

Therefore, if mv = 1

and t = 1, we have F = 1.

As a particular case, if m = 1, v=1, t=1, we have F = 1.

26. The force represented by the weight of a gramme varies from place to place. It is the force required to sustain a gramme in vacuo, and would be nil at the earth's centre, where gravity is nil. To compute its amount in dynes at any place where g is known, observe that a mass of 1 gramme falls in vacuo with acceleration g. The force producing this acceleration (namely, the weight of the gramme) must be equal to the product of the mass and acceleration, that is, to g.

The weight (when weight means force) of 1 gramme is therefore g dynes; and the weight of m grammes is mg dynes.

mass.

27. Force is said to be expressed in gravitation-measure when it is expressed as equal to the weight of a given Such specification is inexact unless the value of g is also given. For purposes of accuracy it must always be remembered that the pound, the gramme, etc., are, strictly speaking, units of mass. Such an expression as

a force of 100 tons" must be understood as an abbrevia

tion for " a force equal to the weight [at the locality in question] of 100 tons."

28. The name poundal has recently been given to the unit force based on the pound, foot, and second; that is, the force which, acting on a pound for a second, gene

rates a velocity of a foot per second. It is

of the

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weight of a pound, g denoting the acceleration due to gravity expressed in foot-second units, which is about 32.2 in Great Britain.

To compare the poundal with the dyne, let x denote the number of dynes in a poundal; then we have

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29. The C.G.S. unit of work is called the erg. It is the amount of work done by a dyne working through a distance of a centimetre.

The C.G.S. unit of energy is also the erg, energy being measured by the amount of work which it represents.

30. To establish a rule for computing the kinetic energy (or energy due to the motion) of a given mass moving with a given velocity, it is sufficient to consider the case of a body falling in vacuo.

When a body of m grammes falls through a height of h centimetres, the working force is the weight of the body —that is, gm dynes, which, multiplied by the distance worked through, gives gmh ergs as the work done. But the velocity acquired is such that v22gh. Hence we have gmh = mv2.

The kinetic energy of a mass of m grammes moving with a velocity of v centimetres per second is therefore mv2 ergs; that is to say, this is the amount of work which would be required to generate the motion of the body, or is the amount of work which the body

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