x )Scales are the reciprocal of each other. e

Transcription

1 9. Reciprocls A Complete Slide Rule Mnul - eville W Young Chpter 9 Further Applictions of the LL scles The LL (e x ) scles nd the corresponding LL 0 (e -x or Exmple : Set the hir line over 4. on the LL 3 scle. 2. Under the hir line red off on the LL 03 scle. x )Scles re the reciprocl of ech other. e Exmple 2: Set the hir line over on the LL 0 scle. 2. Under the hir line red off.07 on the LL scle s the nswer. ote () The reciprocl of ny number from to 22,000 (0-5 to 0 5 for Slide Rules with extended scles) cn be obtined by locting the number on the prticulr e x or e -x scle nd reding off on the corresponding e -x or e x scle s in exmples bove. (b) Even for Slide Rule with LL 0 nd LL 00 scles, there is gp between nd.0009, but this rnge would not often be encountered. For number between 0. nd 6, the LL scles give more ccurcy the CI nd D scles, while outside this rnge (i.e. greter thn 6 nd less then 0.) the CI nd D scles re more ccurte. One of the gretest dvntges of the LL scles for reciprocls, is tht the deciml point is red directly off the scles. e.g. 270 on the LL 3 scle gives on the 270 LL03 scle. Exercise 9() (i) 23 (v) 0.98 (ii) (iii) (iv) (vi) (vii) (viii) Tenth nd Hundredth Powers nd Roots Exmple: Find 2.5 0, , 0, nd. (Fig 9.2) Set the hir line over 2.5 on the LL 2 scle. Under the hir line red off 2. 9,500 on the LL 3 scle s the vlue of on the LL scle s the vlue for on the LL 3 scle s the vlue for

2 A Complete Slide Rule Mnul - eville W Young on the LL 0 scle s the vlue for Further Exmples: For on the LL scle s shown, the other LL scles give LL 3 LL 2 LL LL 0 LL 00 LL 0 LL 02 LL Exercise 9(b) (i).08 0 (ii) (iii) 0.08 (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii) (xiii) (xiv) (xv).7 0 (xvi) ote: () There is no difficulty with locting deciml points, s ll vlues re red off the prticulr LL scle. (b) Recll Positive umbers to Any Powers The LL scles cn be used to obtin for > 0 nd, ny power (positive, negtive or frctionl). The procedure is s follows: For on the LL scle, the D scle give ln (see unit 8). If we multiply this vlue of ln on the 9-2

3 A Complete Slide Rule Mnul - eville W Young D scle by (using the C nd D scle) we then obtin ln (i.e. ln ) on the D scle. If the multipliction ws done using the left index of the C scle (see exmple below), the vlue of is red off (i) The originl LL scle for < < 0. (ii) The LL scle bove the originl for 0 < < 00 (e.g. on the LL 2 then on the LL 3 or on LL 02 then on LL 03 ). (iii) The LL scle below the orginl for 0. < <. (e.g. on the LL 2 then on the LL or on LL 02 then on LL 0 ). In multiplying by ln, on the C nd D scles, if the right index of the C scle is used, the nswers re found on the LL scle bove the LL scles listed in the foregoing method. (see exmple 2). This method cn be extended to lrger or smller number vlues of by moving two or more LL scles insted of one. Exmple : Find.4 2.5,.4 25, nd Set the hir line over.4 on the LL 2 scle. 2. Plce the left index of the C scle under the hir line. 3. Reset the hir line over 2.5 on the C scle. Under the hir line red off on the LL 2 scle s the vlue for ,500 on the LL 3 scle s the vlue for on the LL scle s the vlue for on the LL 0 scle s the vlue for Exmple 2: Find , , , , Set the hir line over 0.6 on the LL 2 scle. 2. Plce the right index of the C scle under the hir line. 3. Reset the hir line over 0.65 on the C scle. Under the hir line red off on the LL 03 scle s the vlue for on the LL 02 scle s the vlue for on the LL 00 scle s the vlue for on the LL 3 scle s the vlue for on the LL scle s the vlue for ote: () For negtive power we simply trnsfer to the reciprocl LL scle (see 9.) (b) Insted of using the C scle in step 2 nd 3 of the bove Exmples, we could hve used the CF scle. Step 2 would then be Plce the index of the CF scle under the hir line. ote, we must use either the C or CF scle in problem, nd not mix them, otherwise it mens the power is either multiplied or divided by π. (c) For powers of numbers which tke us outside the rnge of the LL scles, one of the following procedures cn be used. (i) x ( ) 2. Use the LL scles to evlute nd then squre the results in the usul wy. (ii) (3.64 x 0) x 0 5. (iii) (iv) x Evlute ech seprtely nd multiply together in the usul wy or x ( ) (.63 x 0 2 ) x x x Evlute the first term using the LL scles. To find the second term, use the L scle s x is equivlent to log 0 x 0.6 (i.e. find 0.6 on the LL scle nd red off 3.98 on the D scle (or W 2 scle for the 2/83) s ) (v) (.2 x 0-2 ) x x x 0-2 Agin, we evlute the first two terms s in (iv) bove. Exercise 9(c) (i) (ii) (iii) (iv) (v) (vi)

4 A Complete Slide Rule Mnul - eville W Young (vii) 79 2 (viii) (ix) (x) 4, (xi) (xii) (xiii) (xiv).0 70 (xv) (xvi) 4 22 (xvii) e 4 (xviii), (xix) (xx) Miscellneous Powers nd Roots of Positive umbers Expressing we cn the LL scle to obtin the root of ny positive number. As previously with powers in 9.3, for on n LL scle the D scle gives ln. If we divide this vlue of ln by, we then obtin ln (i.e. ln ) on the D scle. We must note the loction of the deciml point for so s to decide on which LL scle the vlue of will be found. This is done ccording to the sme rules used in 9.3. Exmple Find 3 6 nd Set the hir line over 6 on the LL 3 scle. 2. Plce the 3 of the C scle under the hir line. 3. Reset the hir line over the right index of the C scle. (ote: nd , thus we will find on the LL 2 scle nd 30 6 on the LL scle.) Under the hir line red off on the LL 2 scle s the vlue for on the LL scle s the vlue for Exmple 2: Find nd Set the hir line over 420 on the LL 3 scle. 2. Plce the 5 of the C scle under the hir line. 3. Reset the hir line over the left index of the C scle. Under the hir line red off on the LL 3 scle s the vlue for on the LL 03 scle s the vlue for ote: If we use the right index of the C scle when we divide by for < < 0 the vlue for will be found on the LL scle below the originl LL scle on which ws locted. While, if 0 < < 00 the vlue for will be found on the second LL scle below the originl one. If we use the left index of the C scle when we divide by, for < < 0 the vlue for will be locted on the sme LL scle on which ws locted. While, if 0 < < 00 the vlue for will be found on the LL scle below the originl one. Some specil powers re given in the following tble. On pproprite LL scle Set the H.L over - Under the H.L plce Reset H.L. over On pproprite LL scle under H.L nswer 9-4

5 A Complete Slide Rule Mnul - eville W Young Index of CI scle on the CI scle n Index B B Index BI BI Index K K Index W W 3 2 Index C CF π Index CF C nπ CI M C M M C C M C M CI M Index C θ S sinθ Exercise 9(d) (i) (ii) (iii) (iv) 7. 8 (v) (vi) (vii) 3.2 (viii) 3.26 (ix) sin (x) 0.75 (xi) (xii) π (xiii) (xiv) (xv) 2, (xvi) Logrithms To Any Bse nd Solving Exponentil Equltions A. Logrithm The LL scles cn be used to obtin logrithms to ny bse, by plcing the left or right index of the C scle over the bse s found on the LL scle. Then for ny number on n LL scle its logrithm to the chosen bse is red off the C scle. Exmple : Log Set the hir line over 6 on the LL 3 scle. 2. Plce the left index of the C scle under the hir line. 3. Reset the hir line over 23.5 on the LL 3 scle. 4. Under the hir line red off.765 on the C scle s the nswer. ote: () We could hve used the CF scle bove, by plcing the index of the CF scle under the hir line in step 2, nd thus reding.765 off the CF scle in step 4. (b) For logrithms of other numbers to bse 6, leve the slide s positioned in step 2 nd reset the hir line over the number on n LL scle. (e.g. log s 2 on the LL 2 scle give on the C scle.) Exmple 2: Find log 2.4, Log 2 23 nd log Set the hir line over 2 on the LL 2 scle. 2. Plce the right index of the C scle under the hir line. 3. Reset the hir line over.4 on the LL 2 scle nd red off on the C scle s the vlue for log

6 A Complete Slide Rule Mnul - eville W Young 4. Reset the hir line over 23 on the LL 3 scle nd red off 4.52 on the C scle s the vlue for log Reset the hir line over 650 on the LL 3 scle nd red off 9.34 on the C scle s the vlue for log B. Solving exponentil equtions Becuse Log b x is equivlent to b x the bove problems re the sme s solving n exponentil eqution for n unknown power. Exmple could hve been stted Find x for 6x Thus to solve for n unknown power or exponent we could either express the eqution in logrithmic form or leving it s n exponentil eqution proceed s follows for b x.. Set the hir line over the bse b on the pproprite LL scle. 2. Plce the left or right index of the C scle under the hir line. 3. Reset the hir line over the number on its pproprite LL scle. 4. Under the hir line red off the vlue for x on the C scle nd locte the deciml point ccording to the LL scles used. ote: () To solve the eqution b x for x, we follow the sme method s outlined bove, except in step 4 the vlue of x is red off the CI scle. (b) To solve the eqution b kx for x, the following method cn be used.4 x Exmple: e 9. Set the hir line over e on the LL 3 scle. 2. Plce the.4 of the CIF scle under the hir line. (We cn lso use CI scle here if suitble). 3. Reset the hir line over 9 on the LL 3 scle. 4. Under the hir line red off.57 on the CF scle s the vlue for x. (When the CI scle is used in 2, the nswer in 4 will be on the C scle.) Exercise 9(e) (i) log 5.9 (ii) log 8 24 (iii) log 2 57 (iv) log Find x in the Following: (ix) 5 x 30 (x) 2 x 76 (xi) 5 x 3.5 (xii) 456 x 2 (xiii) 0.3 x 0.95 (xiv) 0.6 x (v) log.5 2 (vi) log 5.3 (vii) log (viii) log x (xv) (xvi) x 6 e 48 (xvii) e πx 64 (xviii) e -3x

7 A Complete Slide Rule Mnul - eville W Young 9-7