The value of the quantity, which is the ratio of first unit-length to the second, is called the change-r from the first system to the second. It is the factor which the numerical value of the quantity in the f system must be multiplied in order to obtain its num cal value in the second system. In the case un consideration

7. Velocity.-We shall next consider how measure of a velocity changes when the units by wh it is measured are changed (using, as above, thick let to represent units, and italics to represent the conc quantities). Let V denote the unit of velocity ba upon Land T as the units of length and time, V' unit of velocity in a second system in which the unit length and time are L' and T'; and let v deno concrete velocity such that a space / is described in time t.

If " denote the measure (or numerical value) of

d and We ength

f the

ratio

or by first

merinder

velocity in terms of the unit V, and ' its value in terms of the second unit V', then

Now the measure of a velocity is the number of units of space ( ́) described in unit time,

the

ich

ers

ete

ed

he

of

a

he

is

Equation (3) enables us to find the measure (n') of the velocity in the second system, when the relations between the fundamental units L and L', T and T'are known. Comparing it with the equation

we see that the unit of velocity varies directly as the unit of length, and inversely as the unit of time. This is usually expressed by saying that the dimensions of the unit of velocity are of one degree in length, and minus one degree in time; or that the dimensions of velocity are or LT-1.

T

It should be noticed that is not a number, ratio in the strict Euclidean sense; it would, perh be better to write it in the form L/T, the solidus or m /standing for the word per. It is in this sense that symbol is used: it indicates that in measurin velocity we have to divide (not in the usual, but i more extended sense) a length by a time. therefore, we write

we mean that the unit of velocity (V) is such that space L is described per time T.

8. Acceleration.-Proceeding with this reason we shall find that in acceleration time is involved tw For acceleration is measured by the increase of velo per unit time; so that if A denote the unit of acce ation, A is equal to V per T, or = V/T, and since have already seen that

in a

hen,

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

a=nA=n'A'

(5)

eler- we

as

s of

em,

ond

the

n.

(6)

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 LT-2.

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.

From (6) we have directly

the

L T2

n' = n

of

L'' T2'

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,

9. Force, Work, and Power.-Before proceedin give the dimensions of other derived units in mechan it may be well to point out the considerations wh determine the choice of any new unit based upon fundamental quantities or upon derived units which h already been fixed. We shall take the unit of force our example.

According to the second law of motion, force measured by the change of momentum which it 1 duces, i.e.