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Misura E Strumenti Di Misura

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Misura E Strumenti Di Misura ® MISURA E STRUMENTI DI MISURA (G) MISURA: g = G : grandezza (numero) U(G) U(G) : unita' di misura di G Come si effettua la misura ? diretta MISURA indiretta confronto diretto con unita' di misura DIRETTA: (Grandezze omogenee) con strumento "tarato" es. : termometro, voltmetro ... INDIRETTA: dalla relazione fisica con altre grandezze es. : P = F/S STRUMENTO DI MISURA: dispositivo per determinare g 30 RIVELATORE: sensibile a grandezza da misurare (sollecitazione) TRASDUTTORE: trasforma sollecitazione STRUMENTO in grandezza facilmente misurabile VISUALIZZATORE: visualizza grandezza trasformata RIVELATORE: Mercurio Es. TERMOMETRO TRASDUTTORE: Mercurio + capillare VISUALIZZATORE: capillare + scala tarata RIVELATORE: bobina mobile in campo magnetico Es. AMPEROMETRO TRASDUTTORE: movimento bobina (rotazione ago) VISUALIZZATORE: posizione ago su scala "tarata" 31 CARATTERISTICHE STRUMENTO INTERVALLO FUNZ. (valore min-max grandezza) PRONTEZZA: tempo necessario per reagire alla sollecitazione τ τ = tempo caratteristico strumento τ (termometro) ~ secondi τ (oscilloscopio) ~ 10 -9 s SENSIBILITA`: ∆R(G) dR R(G): risposta S = = ∆V(G) dV V(G): valore grandezza se R lineare in V(G): S = cost. Altrimenti S funzione di V(G) (Ohmetro) Dimensioni : [G] -1 Es. Bilancia: S in DIV/mg 32 PRECISIONE: capacita' strumento a dare stessa risposta a stessa sollecitazione MISURE RIPETUTE (sensibilit : 1/∆g) N N g(G) g(G) ∆g GIUSTEZZA: assenza di effetti sistematici Difetto strumento (termometro: capillare a sezione variabile) Uso strumento in condizioni errate (regolo tarato a temperatura diversa da temperatura uso) 33 Sensibilit e precisione sono correlate Nel costruire uno strumento occorre bilanciare sensibilita' e precisione precisione (inutile) necessit di ripetere molte errore di misure e fare analisi statistica sensibilit giusto preciso non giusto 33a ERRORI di MISURA Misure ripetute stessa grandezza con stesso strumento nelle stesse condizioni NON danno stesso risultato precisione finita strumento impossibilit controllo tutte condizioni ⇓ incertezze (errori) di misura rispetto a valore vero della grandezza errore da precisione finita strumento > errore sensibilita' ~ incertezze intrinseche misura ⇓ sempre stesso risultato (errore = errore sensibilita') 34 ERRORI CASUALI Valore misurato M(G) differente da valore vero V(G) per quantita' che varia "casualmente" al ripetere della misura ERRORI SISTEMATICI M(G) = V(G) per quantita' che rimane la stessa al ripetere della misura (es. errore di taratura strumento) Vita reale: errori casuali + errori sistematici (buona misura: err. sistematico<< err. casuale) MISURA: migliore stima del valore vero grandezza + valutazione "incertezza" su tale stima = errore "incertezza" ~ V(G) - M(G) g = M(G) +∆[V(G)] errore 35a ERRORI CASUALI Errore di sensibilita' strumento~> fluttuazione intrinseca misure ∆R(G) ∆R(G) 1 S = ∆V(G) = = 1 = ∆V(G) S S 0 a -1 G S = 1 Div mm 0.5 a < V(G) < (a + ∆V(G)) ∆V(G) = 0.5 mm ∆V Miglior stima: M(G) = (a + ) 2 ∆ Errore: + V 2 a = 56.5 mm ∆V = 0.5 mm M(G) = 56.75 + 0.25 mm 36 Errore "statistico" (err.sens.<< fluttuazione misure) n N misure ripetute Istogramma stessa grandezza dei risultati x1, x 2 , ... x i ... xN x ~ 1 S Miglior stima = media aritmetica (da giustificare in futuro) ΣN i x i 1 x = N Errore ∝ larghezza distribuzione risultati misura xmax - x (*) ε errore massimo = min M (semi dispersione massima) 2 ε ε P(x - M < x* < x + M ) deviazione standard = s2 S N Σ 2 i (x i- x) 1 s2 = N - 1 P(x - S < x* < x + S) ~68% (*) errore sensibilita’ un errore massimo 37 ERRORI CASUALI Osservabili solo se strumento di misura e' sufficientemente sensibile Possono essere ridotti con uso di strumenti piu' precisi ma non possono essere mai eliminati completamente Possono essere stimati dato che hanno regolarita' statistiche. Scopo della Teoria degli Errori valutare l'errore presumibilmente commesso nella misura a partire dai dati sperimentali ( misure) DISTRIBUZIONE MISURE: DISTRIBUZIONE DI GAUSS Si dimostra: misure ripetute stessa grandezza fisica (errore sensibilita' << errore statistico) tendono ad avere "distribuzione" rappresentabile con funzione di Gauss (x - xo)2 1 - f(x) = . e 2 σ2 2¹ σ σ 2 due parametri: xo e f(x) σ σ xo − σ xo xo+σ x xo : valore vero della grandezza σ 2 : varianza (larghezza distribuzione) σ ~ 68% misure entro xo ± σ ~ 95% misure entro xo ± 2 σ ~ 99.7% misure entro xo ± 3 (errore massimo 3 σ À) 38a Date N misure stessa grandezza: x ......... x 1 N N Σ i x i 1 miglior stima x : x = o N (media) N Σ 2 i (x i - x) σ 1 miglior stima : S = N - 1 Risultato di misura: σ ) A) 1 misura (assumendo noto x 68% di probabilita' x σ 1 (xM = x1 ± x ) σ xM - xV ² x B) N misure x1 .......... xN N Σ i x i 1 x = N ΣN 2 i (x i - x) σ 1 x = N - 1 σ 68% di probabilita' x = x ± x σ M x N xM - xV ² N 38b Misura T pendolo composto (1 oscillazione/ampiezza ~ 3 gradi) distribuzione di Gauss n A) Cont. sec. mecc. err. sens. = 0.1 s 50 misure Start-stop manuale 10 (s) 2,0 2,2 2,4 x 2,6 2,8 3,0 (x - x)2 1 - f(x) = . e 2σ2 2¹σ n 10 T = 2,337 s 2 distribuzione di Gauss B) Cont. sec. dig. σ = 0,064 s err. sens. = 0.5 10 -2 s 2 Start-stop manuale 100 misure 2,2 2,3 2,4 (s) 2,5 σ: combinazione errore sensibilita' ed errore comando n 20 T 3 = 2,340 s B) Cont. sec. dig. ∆T = 0,005 s err. sens. = 0.5 10 -2 s Start-stop con fotocellula ~ 10 -6 s) 10 20 misure errore comando 2,2 2,3 2,4 (s) 2,5 T 39 CIFRE SIGNIFICATIVE x = 1.23456 u. m. ∆x = 0.231 u. m. Errore: arrotondato ad una (due) cifra significativa 0.154 0.15 0.231 0.2 0.763 0.8 Misura: troncata (arrotondata) al livello dell'errore x = 1.23456 u.m. ∆x = 0.2 x = 1.2 ± 0.2 u.m. x = 1.23456 . 106 u.m. ∆x = 7.6543 . 104 u.m. x = (1.23 ± 0.08) 106 u.m. x = (123 ± 8) 104 u.m. 40 CIFRE SIGNIFICATIVE x = 1.23456 u. m. ∆x = 0.231 u. m. Errore: arrotondato ad una (due) cifra significativa 0.154 0.15 0.231 0.2 0.763 0.8 Misura: troncata (arrotondata) al livello dell'errore x = 1.23456 u.m. ∆x = 0.2 x = 1.2 ± 0.2 u.m. x = 1.23456 . 106 u.m. ∆x = 7.6543 . 104 u.m. x = (1.23 ± 0.08) 106 u.m. x = (123 ± 8) 104 u.m. 40 PROPAGAZIONE DEGLI ERRORI dirette Nelle misure con strumento tarato errore di sensibilita Errore massimo Errore deviazione standard Errore statistico Errore nelle misure indirette (derivate)? G = f(G1, G2, ... Gn) Misurato: G1 ± ∆G1, G2 ±∆G2, G3 ±∆G3 ..... ∆G ? 1. Errori massimi df df + dG = dG1 dG2 + .... dG1 dG2 (variazione infinitesima di G per var.inf. Gi ) a parte termini ordine superiore Passando ai valori finiti: df df + ∆ ∆G = .∆G1 . G2 + ..... dG1 dG2 41 Se voglio massima variazione possibile di G dovro' considerare i valori assoluti delle derivate: df df + ∆ ∆G = . ∆G1 . G2 + ..... dG1 dG2 df n ¶f ∆G = Σ ∆G i i 1 dGi¶Gi Propagazione degli errori massimi. ∆G sara' un errore massimo (rigorosamente valida solo se dipendenza da G i lineare) ∆ Gi piccoli rispetto a Gi 42 ES. 1 y = A. x dy . y = dx x = |A| x T Pendolo: 10 oscillazioni t ± t t 1 . T = 10 T = 10 t ES. 2 x ± x y = x + z z ± z dy . dy . y = dx x + dz z = x + z +1 +1 y = x - z dy . dy . y = dx x + dz z = x + z +1 -1 Misura diametro filo con calibro Palmer: d1 ± d1 do ± do 0 dello strumento d(filo) = d1 - do d1 = d(filo) = d1 + do = d = 2 . dM do = M - 43 - ES. 3 DENSITA’ CUBO m m = V = L3 m ± m (190 ± 1g) L ± L (3.00 ± 0.05 cm) . = ¶m m + ¶L L = . m . = L3 m + -3 L4 L = 1 3m . = L3 m + L4 L ES. 4 y = T2 y = 2T . T ( T2 = 2T . T) ES. 5 g da periodo pendolo l l 2 T = 2 g g = 4 T2 l ± l T ± T g g . g . l l + T T - 44 - 2 g 4 l . T2 l + -2 T3 T = 2 2 4 8 l . = T2 . l + T3 T 4 2 l 4 2l g = T2 . l . l + 2 . T2 . T g g g l T g = l + 2 T ES. 6 Calore specifico con cal. delle mescol. C T’’ - T c = m T - T’ C = ( ca . ma) ± C capacità termica acqua m ± m massa del corpo T’’ ± T” temperatura acqua nel cal. (iniziale) T’ ± T’ temperatura corpo T ± T temperatura di equilibrio - 45 - c 1 T” - T = . C m T - T’ c 1 T” - T m = - m2 . C . T - T’ c C 1 T” = + m . T - T’ c C T” - T T’ = - m . (T - T’)2 c C - (T” - T) - (T - T’) C -T” + T’ = = T m (T - T’)2 m (T - T’)2 c c c = C . C + m . m + . 1 T” - T C T” - T c = m T - T’ . C + m2 T - T’ . m + C 1 C T” - T + m . T - T’ . T” + m (T - T’)2 . ’ C -T” + T’ + m . (T - T’)2 . T (Fare sempre controllo dimensionale) - 46 - 2 - ERRORI STATISTICI G = F(G1 , . , Gn) G1 , . , Gn con errori statistici G1 . Gn Se errori su Gi piccoli rispetto a Gi e Gi statisticamente indipendenti G G ~ 2 . + 2 . + . G G1 G1 G2 G2 1 G 2 G ~ å i .
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