Multiple-Choice Test Runge-Kutta 4 th Order Method Ordinary Differential Equations COMPLETE SOLUTION SET
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1 Multpl-Co Tst Rung-Kutta t Ordr Mtod Ordnar Drntal Equatons COMPLETE SOLUTION SET. To solv t ordnar drntal quaton sn ( Rung-Kutta t ordr mtod ou nd to rwrt t quaton as (A sn ( (B ( sn ( (C os ( (D sn ( Soluton T orrt answr s (B sn ( s rwrttn as sn ( sn ( (
2 . Gvn sn (. Rung-Kutta t ordr mtod s most narl (A. (B 97. (C. (D.8898 Soluton T orrt answr s (C sn (. s rwrttn as ( sn (. t Rung-Kutta t ordr mtod s ( wr ( ( or. ( (. ( sn. (.8.. (.. ( sn. (. and usng a stp sz o..8. t valu o (. 9 usng
3 (.. ( sn. ( (...7. (..79 ( sn. ( ( (.... ( or.. ( (.. ( sn. ( ( ( sn.7 (
4 ... (.7.9 ( sn.7 ( ( ( (.9. ( sn.9 (.. ( (..9. ( (.
5 Rung-Kutta t ordr mtod s most narl (A -. (B -.78 (C -.8 (D.77. Gvn (. Soluton T orrt answr s (A (. s rwrttn as ( (. t Rung-Kutta t ordr mtod s ( wr ( ( or. ( (.. ( ( (..87. ( (.87.. and usng a stp sz o t st stmat o (.9
6 .. (..97. ( ( (..8. (.8.9 ( (... ( or.. ( (... (..... ( ( (
7 ... ( ( ( ( ( ( ( (.8.89 ( ( ( ( ( ( ( ( (.
8 . T vlot (m/s o a parautst s gvn as a unton o tm (sonds ν ( t.8 tan(.7t t Usng Rung-Kutta t ordr mtod wt a stp sz o sonds t dstan travld t o rom t to t sonds s stmatd most narl as (A. m (B 8.97 m (C 9. m (D 7. m Soluton T orrt answr s (C ( t.8 tan(.7t t ν t Rung-Kutta t ordr mtod s S S ( wr ( t S t S t S ( t S or t S ( t S (.8 tan( t (..8 (.7..8 tan.99 t S S
9 ( (.7..8 tan.99 ( t S (.99 ( tan(.7 7. S S ( t t 7 or t 7 S 7. ( t S 77. ( (.8 tan(.7 7. t 7 ( ( tan. S 7. t S ( ( tan...
10 S ( t S ( (. (.7.8 tan.9 S ( m t t 7 ( T dstan travld rom t to t s Dstan travld S S S S ( t S( t ( S( 9. 9.m
11 . Rung-Kutta mtod an drvd rom usng rst tr trms o Talor srs o wrtng t valu o tat s t valu o at n trms o and all t drvatvs o at. I t plt prsson or t rst v trms o t Talor srs ar osn or t ordnar drntal quaton ( 7 would (A ( (B ( ( ( ( (C ( ( ( ( (D ( ( ( ( Soluton T orrt answr s (B T rst v trms o t Talor srs ar as ollows ( ( ( ( ''' '' '!!! our ordnar drntal quaton ( 7 ( Now sn s a unton o ( ( ( ( ( [ ](
12 ( ( ( ( ( d ( ( [ ]( ( ( 8 ( ( ( d ( ( [ ]( ( 8 ( T t ordr ormula or t aov ampl would ( ( ( ( ''' '' '!!! ( ( ( (
13 . A ot sold lndr s mmrsd n an ool ol at as part o a qunng pross. Ts pross mas t tmpratur o t lndr θ and t at θ ang wt tm. I t ntal tmpratur o t ar and t ol at s gvn as C and 7 C rsptvl and Lngt o lndr m Radus o lndr m g Dnst o lndr 7 m J Sp at o lndr 89 g K W Convton at transr ont J Sp at o ol 9 g K Mass o ol g t oupld ordnar drntal quaton gvng t at transr ar gvn m K Ol Clndr (A (B (C (B θ θ θ θ θ θ θ θ θ θ θ θ θ θ
14 Soluton T orrt answr s (A For t lndr t rat o at lost du to onvton ( A( θ θ θ. wr (θ t onvtv oolng ont W/(m -K and s a unton o tmpratur A sura ara o t lndr T nrg stord n t mass s gvn Enrg stord mass m.c.θ wr m mass o t lndr g C sp at o t lndr J/(g-K From an nrg alan Rat at w at s gand - Rat at w at s lost Rat at w at s stord gvs ( θ A( θ θ mc wr W ( θ m K A πr π m ρv πrl (.. m g 7 m 7 π π.. ( πr L (. m..9g Tus ( θ A( θ θ mc.( θ θ θ θ Smlarl or t ol
15 ( θ A( θ θ mc.( θ θ 9.8 θ θ
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