Page:Machinery's Handbook, (6th Edition, 1924, machineryshandbo00indu).pdf/108

Transposition of Formulas

A formula is a rule for the calculation expressed by using letters and signs instead of writing out the rule in words; by this means it is possible to condense, in a very small space, the essentials of long and cumbersome rules. The letters used in formulas simply stand in place of the figures which are to be substituted when solving a specific problem.

An important method for facilitating the use of formulas is known as transposition. As an example, the formula for the horsepower transmitted by belting may be written:

${\mbox{H.P.}}={\frac {SVW}{33,000}}$

in which

H. P. = horsepower transmitted

S = working stress of belt per inch of width, in pounds;

V = velocity of belt in feet per minute;

W = width of belt in inches.

If the working stress S, the velocity V, and the width W, are known, the horsepower can be found directly from this formula by inserting the given values. Assume S = 33; V = 600; and W = 5. Then:

${\mbox{H.P.}}={\frac {33\times 600\times 5}{33,000}}=3$

.

Assume, however, that the horsepower, the stress S, and the velocity V are known, and that the width of belt, W, is to be found. The formula must then be transposed so that the symbol W will be on one side of the equals sign and all the known quantities on the other. The transposed formula is as follows:

${\frac {{\mbox{H.P.}}\times 33,000}{SV}}=W$

.

The quantities (S and V) that were in the numerator on the right side of the equals sign are transposed to the denominator on the left side, and "33,000" which was in the denominator on the right side of the equals sign is transposed to the numerator on the other side. This is in conformity with the general rule for transposition. Symbols which are not part of a fraction, like "H.P." in the formula first given, are to be considered as being numerators (having the denominator 1).

According to the rule given, any formula of the form $A={\frac {B}{A}}$ can be transposed as below:

$A\times C=B$. and $C={\frac {B}{A}}$

Suppose a formula to be of the form:

$A={\frac {B\times C}{D}}$

Then:

$D={\frac {B\times C}{A}};{\frac {A\times D}{C}}=B;{\frac {A\times D}{B}}=C.$

The method given is only directly applicable when all the quantities in the numerator are standing independently or are factors of a product. If connected by + or 0 - signs, the transposition can be made by the method shown only when the whole sum or differences is transposed.

Example:

$A={\frac {B+C}{D}}$; then $D={\frac {B+C}{A}}$; and $A\times D=B+C.$