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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