CUBIC-FOOT VOLUME OF A LOG

Transcription

1 CUBIC-FOOT VOLUME OF A LOG Wys to clculte cuic foot volume ) xylometer: tu of wter sumerge tree or log in wter nd find volume of wter displced. ) grphic: exmple: log length = 4 feet, ech section feet in length, totl numer of cross-sections = 7 Section Cuts Di (inches) Cross-Setionl re (sq. ft.) AREA (sq. ft.) For our exmple, sq. inch on grph = cuic feet. LENGTH (ft.) Use dot grid or plnimeter to get the squre inches of the grphed log. Then, the cuic foot of the log = sq. inches from dot grid * cuic feet

2 ) nlytic: Cone: Volume = / * A * H CONE Cylinder: Volume = A * H PARABOLOID Height Neiloid: Volume = ¼ * A * H NEILOID Proloid: Volume = ½ *A * H CLINDER How re these volume equtions derived? Rotte the line/curve round the xis k :Cylinder / k * :Proloid

3 k * :Cone / k * :Neiloid

4 Derive Prolic Volume Eqution / k * r V k k d k * k * length A* k * d Here, the re of ech slice is, Are *r nd, / Therefore, Are * k * k / r k * Frustrum of cones, cylinders, etc. A frustrum is piece of something, like cone, cylinder, etc. In the derivtion, the integrl ecomes: FRUSTRUM A Am Au 4

5 Frustrum Formule Proloids:.) Smlin s: V A Au L.) Huer s: V Am* L Smlin s nd Huer s re exctly equl if the log is true proloid Cones: V A A*Au Au L Neiloids: V A A *Au A*Au AuL 4 Newton s: This is the most ccurte formul, nd it holds for neiloids, cones, nd proloids V A 4Am Au L 6 Comprison of Formule feet Smlin Overestimtes Huer Underestimtes Di=6 Neiloid Underestimtes Di= Di=8 These re stright lines; conic section Rememer to use inside rk dimeters (di) to clculte the res to use in the volume formuls. 5

6 Tper is the chnge in dimeter with length. For our exmple, 6" 8" Tper 4 tper in 6 ' A =.5454*D =.4 sq. ft. Au =.5 sq. ft. So, V A A*Au Au L.4 Cone: V A Au L.4.5 *' 8 ft Smlin s:.4*.5.5 *' 6ft Huer s: V Am*L.79ft *' 5ft V A 4Am Au 6 L.4 4*.79.5 *' 6ft 6 Newton s: Cord Volume A cord is unit of volume tht equls 8 cuic feet. This is equl to stck of wood with the dimensions 4 x 4 x 8 feet, which includes the wood, rk, nd empty spce. This is common mesure of volume for pulpwood in the South. Avery nd Burkhrt on pge 57 list some common specifictions for pulpwood. Another mesure of volume for pulpwood tht you my encounter in the South is cunit, which equls cuic feet of solid wood. Cunits re not commonly used tody. 6

7 TREE FORM.) Tper Equtions (trnsition to mesuring individul tree content) Mthemticlly descries the stem profile: d DBH f h H d Height Thus, reltive dimeter nd reltive height re relted. h Tper equtions cn e used to cut stnding tree into logs to find their individul volumes DBH Exmple: Let s ssume tht the cross-sectionl re of stem is linerly relted to the height. A (,) (4.5,) H h = 4.5 ft. (H,) 7

8 8.) For given tree, only the re of the se,, is known..) Find the slope, : 4.5 H 4.5 H.) Find the expression for : * 4.5 * 4.5 * In Generl, = BA t DBH H = totl tree height A = cross-sectionl re t h h = height on ole t d d = dimeter t h 4.5 H *4.5 Since we now know tht A = + *h =.5454*d, we cn find expressions for d nd h:.5454 * h d.5454*d h We cn lso find the volume eqution y integrtion: Adh V

9 Let s look t numericl exmple Let = ft nd H = 4.5 feet. Then, * 4.5. Then, A =.45.*h Now, nswer some questions:.) Estimte the dimeter (d in inches) t h = 5 ft. A =.45.(5 ) =.545 ft A =.5454*d.545 =.5454*d d = d = or inches.) Estimte h where d = 4 inches A =.5454*d =.5454*4 =.45.*h.87 =.45.*h.*h = h = 95.8 feet.) Estimte volume to d = 4 inch top: V.45.* hdh 54.ft 4.) Cut tree into 6 foot logs (ssume 6 inch trim llownce nd foot stump). 9

10 Form Fctors & uotients A form fctor is the rtio of ctul tree volume to the volume of specific geometric solid. Since upper-stem dimeters re necessry to compute these volumes, stem form is usully descried y the dimeters themselves. This expression is clled form quotient, which is the rtio of n upper-stem dimeter to DBH. A common form quotient used in mny regions is Girrd form clss. Girrd form clss is the rtio etween the dimeter inside rk t the top of the first 6 foot log nd DBH. DIB7. Girrd formclss DBH