The Planimeter shown by Fig. 26 is the instrument in a form adaptable to various scales, but does not possess any very marked advantages over the simpler form for the purposes of the naval architect or marine engineer, so that notice of it must be brief. In this form of the instrument the unit can be changed by altering the length of the arm which carries the tracer to any of the scales for which the instrument may be made available, and which are found divided upon the variable arm. The scales which are usually provided for are as follows:—
| 10 sq. in. | = 10 square inches | } | ||
| 0·1 sq. f. | = 0·1 square foot | } | ||
| 1 sq. dcm. | = one square decimetre | } | Every total | |
| 0·5 sq. dcm. | = 0·5 square decimetre | } | rotation of | |
| 2000 sq. m. | } | = 2000 square metres on a | } | the roller. |
| 1 : 500 | } | scale 1 : 500 | } | |
| 1000 sq. m. | } | = 1000 square metres | } | |
| 1 : 500 | } | scale 1 : 500 | } |
Describing the simple planimeter more in detail, and referring to Fig. 25, it may be said the outline of the figure to be dealt with is travelled round by a pointer attached to a bar moving on a vertical axis carried by another bar, which latter turns on a needle point slightly pressed into the drawing surface. The bar with the pointer is provided with a revolving drum having a graduated circumference and a disc counting its revolutions. The drum is divided into 100 parts, reading into a vernier, which gives the reading of the drum’s revolution to the 1/1000 part of its circumference. Upon the same axis as the drum an endless screw is cut, working into a worm wheel of ten teeth connected with the counting disc, which records the revolutions of the drum.
To use the planimeter, place the instrument upon the paper so that the tracing point, roller, and needle point, all touch the surface at any convenient position. Press the needle point down gently, so that it just enters the paper, and place the small weight supplied with the instrument over it. Make a mark at any part of the outline of the figure to be computed, and set the tracing point to it. Before commencing read off the counting wheel and the index roller. Suppose the counting wheel marks 2, the roller index 91, and the vernier 5, then, the unit in this case being 10 sq. ins., write this down 29·15 (for the proportional or variable-scale planimeter this reading would be 2·915.) Follow with the tracing point exactly the outline of the figure to be measured in the direction of the movement of the hands of a watch, until you arrive at the starting point; now read the instrument. Suppose this reading to be 47·67, then by deducting the first reading (29·15) the remainder (18·52) indicates that the measured area contains 18·52 units—i.e., square inches—which is the final result, so far as the instrument is concerned. To obtain the actual area in feet, however, this result must be multiplied by the number before explained corresponding to the scale on which the figure that has been measured is drawn.[35] Assuming the scale to have been ¼-inch per foot, then 18·52 inches multiplied by 16—the appropriate multiplier for that scale—gives 296·32 square feet, the exact area.
Several important points remain to be noticed in connection with the use of the instrument. As a rule, the areas to be measured in connection with ship designing are on a small scale, and the fixed or needle point about which the instrument moves can always be placed outside the figure measured, in which case the process remains as above stated. It should be mentioned, however, that by placing the needle point inside the figure, in such a position as to enable the operator to follow its contour a larger figure can be measured at one operation—the reading, however, being less than the true area by a constant number which varies slightly with the construction of each instrument, and which is found engraved on the small weight already referred to (on the top of the bar in the proportional planimeter). Adding this constant number to any reading taken by the instrument placed as described, gives the true area.
The counting disc may go through more than one revolution forwards or backwards. If the needle point be outside the figure traversed the counting disc can only move forwards (as 9, 0, 1, 2, &c.): that is, provided the figure has been traced in the manner directed—in the direction of the hands of a watch. Then as many times as the zero mark passes the index line add 10·000 to the second reading. If the needle point be inside the figure, the disc can move either forwards or backwards. If moving backwards, as 2, 1, 0, 9, &c., then add 10·000 to the first reading.
Before passing from the subject of the planimeter it may be both interesting and useful to give an example of a calculation involving its use. Subjoined is a specimen displacement and longitudinal centre of buoyancy calculation, and any one familiar with the prodigious array of columns and figures pertaining to a “displacement sheet” of the ordinary kind cannot fail to appreciate the advantages of the specimen, both with respect to simplicity of arrangement and curtailment of the amount of calculation ordinarily involved:—
EXAMPLE OF SHIP DISPLACEMENT, WORKED OUT BY PLANIMETER.
| No. of Sections for Displacement. | Area of Half Sections. | Simpson’s Multipliers. | Functions. | Multipliers for Centre of Buoyancy. | Moments for Centre of Buoyancy. | |
|---|---|---|---|---|---|---|
| Successive Readings of Planimeter. | Difference between Readings = Area in sq. ins. | |||||
| 52·73 | ||||||
| 1 | 52·73 | 0·0 | 1 | 0·0 | 0 | 0·00 |
| 2 | 54·55 | 1·82 | 4 | 7·28 | 1 | 7·28 |
| 3 | 58·98 | 4·43 | 2 | 8·86 | 2 | 17·72 |
| 4 | 64·61 | 5·63 | 4 | 22·25 | 3 | 67·56 |
| 5 | 70·73 | 6·12 | 2 | 12·24 | 4 | 48·96 |
| 6 | 77·05 | 6·32 | 4 | 25·28 | 5 | 126·40 |
| 7 | 83·37 | 6·32 | 2 | 12·64 | 6 | 75·84 |
| 8 | 89·64 | 6·27 | 4 | 25·08 | 7 | 175·56 |
| 9 | 95·75 | 6·11 | 2 | 12·22 | 8 | 97·76 |
| 10 | 01·45 | 5·7 | 4 | 22·8 | 9 | 205·20 |
| 11 | 06·09 | 4·64 | 2 | 9·28 | 10 | 92·80 |
| 12 | 08·57 | 2·48 | 4 | 9·92 | 11 | 109·12 |
| 13 | 08·57 | 0·0 | 1 | 0·0 | 12 | 0·00 |
| (Com. int.) (mult. for ¼th scale) (both sides) | } | 168·12 | |||
| 28·6 × 16 × 2 | } | = 8·716 | 168·12 ) | 1024·20 | |
| (Simpson’s Mult.) (cub. ft. to ton.) | } | 6·09* | |||
| 3 × 35 | } | 100872 | |||
| 16812 | |||||
| 117684 | |||||
| * 6·09 × 28·6 (Com. Int.) | 134496 | ||||
| = 174·2 Centre of Buoy. | |||||
| forward of No. 1 Ordinate. | 1465·33392 tons m’l’d dis’p’t. | ||||