Physics and Measurement
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Transcript Physics and Measurement
PHYSICS AND MEASUREMENT
FUNDAMENTAL QUANTITIES
SI units:
Time – second
Mass – Kilogram
Length – meter
TIME
Before 1967, a second was defined as (1/60)(1/60)(1/24)
of a mean solar day. As this is based on the rotation of
Earth, it is not universal.
Redefined as 9,192,631,770 times the period of vibration
of radiation from the cesium-133 atom making use of
the high precision atomic clock.
MASS
Defined by the mass of a specific platinum-iridium alloy
cylinder kept at the International Bureau of Weights and
Measures in France.
Established in 1887
Duplicate at the National Institute of Standards and
Technology in Gaithersburg, MD.
LENGTH
A meter is the distance traveled by light in a vacuum
during a time of 1/299,792,458 second. (1983)
Originally defined as one ten-millionth of the distance
from the equator to the North Pole along a longitudinal
line that passes through Paris. (1799, earth-based)
Until 1960, distance between to marks on a specific
platinum-iridium bar.
Between 1960-1970, defined as 1,650,763.73
wavelengths of orange-red light emitted from a
krypton-86 lamp.
PERCENT ERROR
Percent error is a way of comparing a calculation or a
measurement to an exact, known value.
STANDARD DEVIATION
The standard deviation is one number that is used to
express how far (on average) the data points are from
the mean value of the data set.
𝜎=
𝑥−𝑥
𝑛−1
2
ADDING AND SUBTRACTING
WITH STANDARD DEVIATION
Addition
1. Add the principal numbers.
2. Add the standard deviations
(𝑥 ± 𝜎𝑥 ) + (y ± 𝜎𝑦 ) = (𝑥 + y) ± (𝜎𝑥 +𝜎𝑦 )
Subtraction
1. Subtract the principal numbers.
2. Add the standard deviations
(𝑥 ± 𝜎𝑥 ) - (y ± 𝜎𝑦 ) = (𝑥 - y) ± (𝜎𝑥 +𝜎𝑦 )
MULTIPLYING AND DIVIDING
WITH STANDARD DEVIATION
Multiplication
1. Multiply the principal numbers
𝑥∗𝑦
2. Determine the fractional uncertainty (FUN) of each principal number.
𝐹𝑈𝑁 𝑥 =
𝜎𝑥
𝑥
and 𝐹𝑈𝑁 𝑦 =
𝜎𝑦
𝑦
3. Add the fractional uncertainties to get total FUN.
FUN (total) = FUN (x) + FUN (y)
4. Multiply total FUN by the principal result to get the total uncertainty.
𝜎𝑡𝑜𝑡𝑎𝑙 = 𝑥 ∗ 𝑦 ∗ 𝐹𝑈𝑁 (𝑡𝑜𝑡𝑎𝑙)
EXAMPLE CALCULATION
Suppose in a lab situation that you would like to calculate the velocity of an object
with standard deviation. The displacement of the cart based on your
measurements is (1.57 +/- 0.07) meters and the cart’s time to travel this distance is
(0.68 +/- 0.02) seconds.
To calculate velocity: 𝑣 =
∆𝑥
∆𝑡
=
(1.57 +/− 0.07) meters
(0.68 +/− 0.02) seconds
To get your principle avg velocity, divide your principle numbers.
∆𝑥
1.57 meters
𝒗=
=
= 𝟐. 𝟑𝟏 𝒎/𝒔
∆𝑡 0.68 seconds
EXAMPLE CALCULATION CONT…
To get your standard deviation,
First find your fractional uncertainty for each value (essentially your percent
uncertainty, in decimal form)
𝑭𝑼𝑵 ∆𝒙 =
𝜎∆𝑥
∆𝑥
=
0.07
1.57
= 𝟎. 𝟎𝟒𝟓 and 𝑭𝑼𝑵 ∆𝒕 =
𝜎∆𝑡
∆𝑡
=
0.02
0.68
= 𝟎. 𝟎𝟐𝟗
Next, add your fractional uncertainties to get total uncertainty
𝐹𝑈𝑁 𝑡𝑜𝑡𝑎𝑙 = 𝐹𝑈𝑁 ∆𝑥 + 𝐹𝑈𝑁 ∆𝑡 = 0.045 + 0.029 = 𝟎. 𝟎𝟕𝟒
Finally, multiple your total fractional uncertainty to your principle number to get your
total standard deviation.
𝑚
𝒎
𝝈𝒗 = 𝒗 ∗ 𝐹𝑈𝑁 𝑡𝑜𝑡𝑎𝑙 = 2.31 ∗ 0.074 = 𝟎. 𝟏𝟕
𝑠
𝒔
So… 𝒗 = 𝟐. 𝟑𝟏 ± 𝟎. 𝟏𝟕
𝒎
𝒔