Mesure et incertitudes - Un cours de physique en spéciale PC ...
Mesure et incertitudes - Un cours de physique en spéciale PC ...
Mesure et incertitudes - Un cours de physique en spéciale PC ...
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
⊲ <br />
⊲ <br />
⊲ <br />
⊲ <br />
<br />
<br />
X = mesuran<strong>de</strong>,<br />
<strong>de</strong> valeur vraie x 0<br />
à déterminer<br />
qui peut être<br />
directe ou indirecte via<br />
<strong>de</strong>s capteurs,<br />
<strong>de</strong>s instrum<strong>en</strong>ts <strong>de</strong> mesure,<br />
<strong>de</strong>s mesurages d'autres gran<strong>de</strong>urs, ...<br />
Représ<strong>en</strong>tation d'une gran<strong>de</strong>ur<br />
X <strong>en</strong> sci<strong>en</strong>ces <strong>physique</strong>s<br />
X = x +/- U(x) [X]<br />
valeurs obt<strong>en</strong>ues par le mesurage<br />
(<strong>en</strong>semble <strong>de</strong>s processus <strong>de</strong> mesure)<br />
x = mesure<br />
incertitu<strong>de</strong><br />
sur x<br />
<br />
unité <strong>de</strong> X<br />
cf chapitre :<br />
Invariance dim<strong>en</strong>sionnelle<br />
qui peut être une<br />
- incertitu<strong>de</strong> type, notée u(x), caractérisant<br />
la dispersion <strong>de</strong>s résultats <strong>de</strong> la mesure,<br />
- incertitu<strong>de</strong> élargie, notée U(x) définissant<br />
un intervalle cont<strong>en</strong>ant, à un niveau<br />
<strong>de</strong> confiance donnée, la gran<strong>de</strong>ur mesurée.<br />
0
= 0 <br />
<br />
ε = |x − x0|<br />
= <br />
<br />
εr =<br />
|x − x0|<br />
|x|<br />
<br />
0 <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
= <br />
<br />
<br />
<br />
<br />
<br />
<br />
X = x0 + ε + ∆<br />
0 ε <br />
∆ <br />
<br />
<br />
<br />
erreur erreur<br />
systématique aléatoire<br />
e<br />
D<br />
x 0 xi<br />
valeur vraie (inconnue) une mesure<br />
moy<strong>en</strong>ne (inconnue)<br />
d'une infinité <strong>de</strong> mesures
ε ∆ <br />
E(X) = x0<br />
<br />
u(x) <br />
<br />
U(x) <br />
k k<br />
<br />
<br />
k 2<br />
[x − U(x); x + U(x)] <br />
<br />
k = 3 <br />
<br />
<br />
<br />
<br />
<br />
<br />
N X x < x ><br />
<br />
〈x〉 ˆ= 1<br />
N ΣN k=1xk<br />
〈s〉 = 1<br />
T<br />
s(t) dt<br />
T 0<br />
〈x2 〉 ˆ= 1<br />
N ΣNk=1x2k <br />
N → ∞ x → µ <br />
σx {x} <br />
<br />
(x 2<br />
σx = − µ) = 〈x2 〉 − µ 2<br />
x <br />
σx 〈x − 〈x〉〉 = 〈x〉 − 〈x〉 = 0 <br />
(x − 〈x〉) 2 √ <br />
x
N < 20 <br />
<br />
<br />
<br />
N<br />
(xi − x)<br />
i=1<br />
σN−1 =<br />
2<br />
N − 1<br />
x σx <br />
<br />
<br />
<br />
N X <br />
xk<br />
σN−1<br />
<br />
<br />
σN−1 <br />
σx <br />
x<br />
<br />
N <br />
<br />
N <br />
σm<br />
<br />
σm <br />
σm = σN−1<br />
√<br />
N<br />
<br />
n x1, x2, · · · , xn<br />
u(x) x = x1+x2+···+xp<br />
p<br />
<br />
u(x) = σn−1<br />
√n<br />
X <br />
N <br />
r = σu(x)<br />
u(x) ≈<br />
<br />
1<br />
(2N − 1)
N = 10, 25 % N = 50 <br />
r ≈ 10 %<br />
<br />
<br />
<br />
<br />
✎<br />
✍<br />
r <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
a b a b <br />
|b − a|<br />
u(x) = <br />
6<br />
a b <br />
|b − a|<br />
u(x) =<br />
2 √ 3 <br />
<br />
|b − a|<br />
u(x) =<br />
2 √ 6 <br />
<br />
X = x0 + E1 + E2 + · · · + En <br />
E1, E2, · · · , En <br />
<br />
u 2 (X) = u 2 (E1) + u 2 (E2) + · · · + u 2 (En)<br />
<br />
☞<br />
✌
f x1 x2 xn <br />
X1 X2 Xn <br />
f <br />
<br />
<br />
f(x1, x2, ..., xn) σxi xi<br />
σf f σ 2 2<br />
<br />
f = Σi <br />
∂f <br />
<br />
σ<br />
∂xi<br />
2 xi <br />
<br />
<br />
f = x ± y ⇒ σf =<br />
√<br />
1 Nσ 2 N<br />
<br />
σ 2 x + σ 2 y<br />
<br />
N X σxi = σ ∀i σf = σ<br />
√ N f = 1<br />
N Σixi ⇒ σf =<br />
f = ax + b ⇒ σf = |a| σx σf = a 2 σ 2 x <br />
<br />
f = x × y<br />
f = x/y<br />
<br />
⇒ σf<br />
|f| =<br />
f = x n × y k ⇒ σf<br />
|f| =<br />
<br />
σx<br />
x<br />
n σx<br />
x<br />
2<br />
2<br />
+<br />
+<br />
2 σy<br />
y<br />
<br />
k σy<br />
2 y<br />
<br />
σf =<br />
<br />
(nx n−1 y k ) 2 σ 2 x + (x n × ky k−1 ) 2 σ 2 y <br />
<br />
f<br />
<br />
f(x1, x2, ..., xn) ∆xi xi<br />
<br />
<br />
∆f f ∆f = Σi <br />
∂f <br />
<br />
∂xi<br />
∆xi <br />
f<br />
f df = Σi ∂f<br />
∂xi dxi <br />
∆f <br />
<br />
<br />
f = x − y ⇒ ∆f = ∆x + ∆y<br />
f = ax + b ⇒ ∆f = |a| ∆x<br />
f = x × y ⇒ ∆f = |y| ∆x|x| ∆y ⇒ ∆f ∆x ∆y<br />
|f| = |x| + |y|<br />
f = xn × yk ⇒ ∆f = nxn−1yk <br />
∆xxn<br />
× kyk−1 ∆f<br />
∆y ⇒<br />
∆x ∆y<br />
|f| = |n| |x| + |k| |y|<br />
<br />
∆ lg (f) = ∆ (n lg (x) + k lg (y)) ⇒ ∆f<br />
f<br />
= |n| ∆x<br />
x<br />
+|k| ∆y<br />
y
k <br />
x U(x) = ku(x), u(x) <br />
X ∈ [x − U(x); x + U(x)] p <br />
k U(x <br />
<br />
k <br />
<br />
<br />
<br />
X m σ <br />
(−σ X − m +σ) ≈ 68 %<br />
(−2σ X − m +2σ) ≈ 95 %<br />
(−3σ X − m +3σ) ≈ 99, 7 %<br />
2σ 3σ 5σ <br />
<br />
En T.P., on pr<strong>en</strong>dra 2σ pour l ′ incertitu<strong>de</strong> élargie <br />
<br />
N <br />
N > 1 σ σN−1 <br />
<br />
N < 10) σN−1 σ <br />
<br />
U(x) = tp,N × σN−1<br />
√ N<br />
tp,N <br />
p <br />
<br />
t0.95,N <br />
<br />
t0.95,N <br />
t ≈ 2
X = x ± U(x) unité.<br />
U(x) <br />
x.<br />
σℓ = 0, 25 ℓmes = (8, 6 ± 0, 5) <br />
ℓmes = (8 ± 0, 5) [7, 5; 8, 5] <br />
[8, 1; 9, 1] <br />
ℓmes = (8, 653 ± 0, 5) 0, 053 8, 653 <br />
<br />
<br />
U <br />
U <br />
<br />
<br />
x = 12, 3257 u(x) 0, 232 k = 2 U = 0, 464. <br />
0, 04 X = 12, 33 ± 0, 47<br />
x = 123, 385 u(x) 2, 892 k = 2 U = 5, 784 <br />
0, 57 X = 123, 4 ± 5, 8.<br />
<br />
<br />
<br />
<br />
<br />
68%<br />
<br />
r <br />
<br />
<br />
<br />
<br />
r = 0, 99<br />
r = 0, 98
m σ > 0 <br />
R f(x) = 1<br />
σ √ 2π exp<br />
<br />
(x − m)2<br />
−<br />
2σ2 <br />
<br />
<br />
Xi <br />
m σ<br />
Xi X = 1<br />
n<br />
Xi <br />
<br />
n <br />
n<br />
i=1<br />
<br />
<br />
<br />
<br />
<br />
<br />
Pn,p(k) n <br />
p k Pk = C k np k (1 − p) n−k C k n!<br />
n =<br />
k!(n − k)!<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
N <br />
p ≪ 1 n <br />
n ≪ N Pλ(n) = λn<br />
n! e−λ , où λ = Np<br />
n = Np = λ σ = √ λ