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But since the quantity measured in both systems is the same, we have,

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A comparison of equations (5) and (6) shows that the unit of acceleration varies directly as the unit of length and inversely as the square of the unit of time; in other words, that its dimensions are or LTTM3.

The following example will illustrate the way in which these equations are applied :

Ex. 1. Express the acceleration due to gravity in terms of the mile and the hour as the units of length and time, its value being 32 when the foot is the unit of length, and the second the unit of time.

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an equation which gives us the required measure (n') in the new system, when the relations between the fundamental units in the old and new systems are known. In the example n =

32,

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Before proceeding to

9. Force, Work, and Power.

give the dimensions of other derived units in mechanics, it may be well to point out the considerations which determine the choice of any new unit based upon the fundamental quantities or upon derived units which have already been fixed. We shall take the unit of force as our example.

According to the second law of motion, force is measured by the change of momentum which it produces, i.e.

fa (rate of change of mv),
:. f∞ ma,

(where a denotes acceleration), or

f=k•ma.

The units of mass and acceleration are already fixed, but we may make the unit of force whatever we please, and it will obviously be most convenient to choose it so that the constant multiplier k shall be equal to unity. Our equation will now become

f=ma.

Now suppose m and a to be each equal to unity; then f will also be equal to unity. Thus our unit of force (F) is defined as being that force which produces unit acceleration in unit mass. We may therefore write

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an equation which gives the dimensions of force.

Since work is measured by the product of force into the distance through which the force acts, the dimensions of work will be those of force multiplied by length, or

W=MLT-2XL=ML2T-2.

The power (or activity) of an agent is measured by the rate at which it does work; hence the dimensions of power are

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Knowing the dimensions of these quantities, we can perform the change of units without going through the lengthy reasoning of §§ 7 and 8; we shall indicate the general method to be followed, but it will be best understood by reference to the actual examples given. [See also equations (5) and (6) in § 8.]

Let 9 be any concrete quantity, and let its measure be n in terms of the unit Q, which is based upon the fundamental units M, L, and T; we wish to find its measure n' in a new system in terms of the unit Q' which is based upon M', L', and T'. Since the quantity meas

ured in both systems is the same,

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Let the dimensions of Q be MLT*; substituting for O and Q' in equation (7) we have

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Ex. 2. Find the number of dynes in a poundal (the poundal being the British absolute unit of force, based upon the pound, foot, and second).

Referring to equations (7) and (8) we see that, since n = 1, the required number n' is the change-ratio or multiplier for changing from British to C.G.S. units of force. The dimensions of force are MLT, so that x= 1, y = I, and z = T and T', the units of time, are the same (one second) in both systems.

- 2.

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Ex. 3. Find the value of a horse-power in watts, a horsepower being equivalent to 550 foot-pounds per second, and the value of g being 32.18.

As the foot-pound is a gravitation unit, we shall first have to reduce to the corresponding absolute unit by multiplying by g—

550 foot-pounds=550 × 32.18 foot-poundals.

The dimensional equation for finding the equivalent rate of working in C.G.S. units (ergs per second) is,

550 × 32.18 × ML2T¬3= 12' × M'L'2T'~3.

The units of time (T and T') are the same in both systems; and, as in Example 2,

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horse-power = 745.8 × 107 ergs per second, or

Thus (since I watt

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10. Magnetic and Electrical Units. The dimensions of the most important of these are given below, and it will be useful practice for the student to deduce them from the corresponding physical laws, as we have done in the preceding articles.

DIMENSIONS OF MAGNETIC UNITS.

Strength of magnetic pole
Magnetic moment of magnet
Strength of magnetic field

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ML1T¬1

MILIT

MILIT

-1

Ex. 4. The dimensions of magnetic intensity (or strength of field) are MLT, and the horizontal intensity of the earth's magnetic force at Aberystwith is 0.1774 in C.G.S. units: what is its value in British (foot-grain-second) units?

The intensity is the same, whatever units we employ to measure it. Let be its numerical value in the British system, in which the unit of field intensity is H', the corresponding unit in the C.G.S. system being H;

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and the units of time (T and T') are the same in both

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