The notes for the physics lab on propagation of uncertainties focus on understanding how measurement uncertainties affect calculated results. Key topics include identifying sources of error, combining uncertainties in addition, subtraction, multiplication, and division, and using significant figure...
topic 1 : propagation of uncertainties
uncertainty : intrinsic er ror i n estimating measurements question 2a
·
·
54 cm
·
question I f dividing uncertainties propagation
I I De = uncertainty
·
convert value (4 . 9170 + 0 . 0005) cm to inches ,
1 inch = 2 .
+War =A
br
↑
. 9170cm
.Finch
4 =
central value = 1 .
as r
e
er ror propagation ↳ for r
·
=
Linch
0 . 0005cm .
W
0 0001969 . 0
0
your measurements affect the results derived from measurements
= =
how the uncertainty in
.
·
the uncertainties propagate into final results
(0 .005000S 23 . .
·
ar = .
(1 9358 = 0 .
.
0002) in
·
er ror due to limitation of equipment
(1 . 222 + 0 006)
.
·
determine uncertainty by smallest value device can measure ·
question 2b
eg : a ruler whose smallest value is I mm will have an uncertainty of 10 S of
.
mm ·
what is the value
= + dz =
dd its
uncertainty ? can make multiple measurements a their results
·
we average
eg : measured a value of 4 95 cm on ruler whose smallest value is 0 . 1 ,
cm uncertainty is 10 05 cm d1 =
(1 000 + 0 005)
our estimate would be the
. .
value
.
mean
.
·
(4 95 = 0 05) cm (0
d2 =
230 = 0 005)
=
.
.
.
.
e
li * a smaller mean more accurately represents the data
·
digital caliper measures to nearest 100th of a mm 0 . 01 mm
minimum : i
eg : caliper measures 49 17mm
, uncertainty = = 0 005 mm
we can calculate the estimate of this uncertainty via standard deviation
.
1. 000 005 995
.
0
·
-
0 .
= .
D1 22
if
↳ you conver t to cm : 4 9170 cm uncertainty = 10 0005cm .
0 230 -0 005
. .
,
.
.
= 0 225 li = value
Or
.
Ill
·
question I a
maximum :
l = mean
·
read 49 17 mm but fluctuates btwn
. 49 Olmm 49 22mm what is the error ? 1 000 + 0 005 = 1 005
Al 6 l
.
,
.
. .
.
,
video practice problems
average error : :
uncertainty
D = AXmin + DX max
0 11 mm 0 0005 + 0 0005 = 0 001
equation : a =
xy +
4x(5y +
5) ,
what is Da ?
0 105mm
.
=
.
.
= .
.
2 in order to find Ad follow PEMDAS
,
~
according to rounding : 0 0010
.
error :
e r ro r :
.
1 start w/ by +
24 treat each term independent i
(49 17 = 0 11) mm
-)
.
010)
.
(1 . 230 + 0 . m
A5y292b(sy +
8-value . next
2 ,
multiplication rule w/4 x
·
if 8 < 1 , the two values agree
·
question 2C adding uncer tainties propagation
·
errors might be too large or overestimated given X A X & Y 1Dy , determine D2 in ter ms of DX & Dy
.
b(4x(Sy +
-
·
if 1 < f < 3 ,
the two values are intension 2 = x +
y . find
3 uncertainty of Xy
D2 =
DX +
DY
·
if 8 < 3 the two values do not agree DXY
,
errors might be too small 4. uncertainty of combining all terms
question 2d
·
subtracting
a is
uncertainties propagation
·
equation : given X A X & Y 1Dy determine Aw in ter ms of DX & Dy
.
-
·
,
Ad =
x(xy + 4x(5y +
Ile-12) w = X Y
& value must be rounded to
-
Se *
same amt. of sigtigs as the value
(Dl1)2 + (De2)
<
AW = Dx +
AY 1
.
uncertainty of F = ma
wl the lowest amt. of sigfigs
a
.
question ze multiplying uncertainties propagation
·
·
rounding uncertainties multiplication rule :
Be Be AWw
DA
D
It would result a value of 1
= ,
·
round uncertainty to one significant figure unless in
A
D DA =
x(xy) = DX -
y +
XDY
W
·
then round your central value to the same decimal place as your rounded uncertainty
DF =
Am .
a + m Da
for 1
.
↳ A = w
eg : (4 225 + 0 0398) .
. p (4 .
23 = 0 . 04)
= 0 1293) (12 09 + 0 13)
eg : (12
093
.
.
a .
.
1 = (2 .
310 1 0 .
005) m w = (1 890 ! 0 005)m
. .
A =
(0 .0050005 .
(2 310
.
-
1890 0 =
(4 37 10 02)m2 . .
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