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