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Electrical Engineering

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Dec 6, 2023

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Measurements and Error Analysis Lab Jaden Ho, Cameron Nguyen October 4, 2023 1 Objective There are a couple of objectives with this lab. We want to use various mea- suring devices and understand the uncertainties for each measuring device. We also want to understand the importance of the error propagation formula, which will help us with labs in the future. 2 Theory The linear approximation helps estimate values at smaller quantities. The error propagation formula helps us analyze any systematic or random errors in our lab. We are bound to make a small error from the theoretical value so it is important to use the error propagation formula. Error propagation formula: σ f = s ( ∂f ∂x ) 2 ∗ σ x + ( ∂f ∂y ) 2 ∗ σ y + ( ∂f ∂z ) 2 ∗ σ z (1) Volume of a table top: v = lvh (2) Where v represents volume and l,v,h are the length, width, and height of the classroom table top. We have to find the partial derivatives to get our error propagation of the volume. ∂v ∂l = wh (3) ∂v ∂w = lh (4) 1

∂v ∂h = lw (5) σ v = r ( ∂v ∂l ) 2 ∗ σ x + ( ∂v ∂w ) 2 ∗ σ y + ( ∂v ∂h ) 2 ∗ σ z (6) Density : ρ = m v (7) Where ρ represents density(g/ cm 3 ), m represents mass(g), and v represents volume( cm 3 ). Using the error propagation formula: σ ρ = r ( ∂ρ ∂m ) 2 ∗ σ 2 m + ( ∂ρ ∂v ) 2 ∗ σ 2 v (8) ∂ρ ∂m = ( m v ) ′ = (1 /v ) ∗ 1 = 1 v (9) ∂ρ ∂v = ( m v ) ′ = m ∗ ( − 1 /v 2 ) = m − v 2 (10) 3 Apparatus 1. Meter stick/ruler: Used to measure length, width, height of the alu- minum block. Used naked eye to determine the closest measurement. 2. Vernier Calipers: There’s two ”jaws” to measure the inner dimensions and outer dimensions of an aluminum block. Measurements are made by closing both jaws and looking at the zero line of the vernier ruler to find a precise millimeter measurement. 3. Micrometer: I put the aluminum block’s height, length, and width into the U-frame of the micrometer and locked the block into place. I then rotated the thimble to find the precise millimeter measurements. 4. Triple Beam Balance: I had to place aluminum block onto the weight and determine the mass of the block. If the block was going down, it meant my measurement was to small and if the block was falling up, 2

the opposite would happen. I kept on doing trial and error until the weight was right on the center line. 5. Digital Balance: I placed the block on to a digital scale and it told me the mass. 4 Procedure 1. Calculate the volume of the aluminum block by using the metric ruler to find the length, width, and height. Multiply all three lengths together. Record data. 2. Calculate the volume of the aluminum block by using the vernier calipers to find the length, width, and height. Multiply all three lengths together. Record data. 3. Calculate the volume of the aluminum block by using the vernier calipers to find the length, width, and height. Multiply all three lengths together. Record data. 4. Calculate the mass of the aluminum block by using the triple beam balance. Use naked eye to see if the block is balanced. 5. Use the digital balance to determine to block’s mass. 6. Use the propagation formula to determine the uncertainty of the den- sity. Find the partial derivatives with respect to mass and volume and plug values into the formula. 7. Find the percent error by dividing (experimental - theoretical) by theoretical. 5 Data Percent errors between calculated densities and expected density. • Length of the table top: 213.5 cm • Width of the table top: 91.5 cm 3

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• Height of the table top: 91.5 cm • Uncertainty of the volume: 1443.39 cm 3 • Volume with metric ruler: 32.4 cm 3 • Length with metric ruler: 7.1 cm • Width with metric ruler: 3.8 cm • Height with metric ruler: 1.2 cm • Volume with vernier calipers: 37.01 cm 3 • Length with vernier calipers: 7.12151 cm • Width with vernier calipers: 3.8573 cm • Height with vernier calipers: 1.2872 cm • Volume with micrometer: 35.81 cm 3 • Length with micrometer: 7.213 cm • Width with micrometer: 3.857 cm • Height with micrometer: 1.287 cm • Mass with triple beam balance: 96.5 g • Mass with digital balance: 96.71 g • Uncertainty of metric ruler: 0.05 cm • Uncertainty of vernier calipers: 0.001 cm • Uncertainty of micrometer: 0.002 mm • Uncertainty of triple beam: 0.05 g • Uncertainty of triple beam: 0.01 g 4

6 Calculations To find the uncertainty of the volume of the table top: σ v = r ( ∂v ∂l ) 2 ∗ σ 2 x + ( ∂v ∂w ) 2 ∗ σ 2 y + ( ∂v ∂h ) 2 ∗ σ 2 z (11) σ v = p (91 . 5 2 ) 2 ∗ 0 . 05 2 + (213 . 5 ∗ 91 . 5) 2 ∗ 0 . 05 2 + (213 . 5 ∗ 91 . 5) 2 ∗ 0 . 05 2 (12) = 1443 . 39 cm 3 ∂ρ ∂m = 1 v (13) ∂ρ ∂v = − m v 2 (14) We have to use the error propogation formula: σ ρ = r ( 1 v ) 2 ∗ σ 2 m + ( − m v 2 ) 2 ∗ σ 2 v (15) Uncertainty of density for triple beam balance and metric ruler: Density = 96 . 5 32 . 4 = 2 . 98 g/cm 3 (16) σ ρ = r ( 1 v ) 2 ∗ 0 . 05 2 + ( − m v 2 ) 2 ∗ 0 . 05 2 (17) σ ρ = r ( 1 32 . 4 ) 2 ∗ 0 . 05 2 + ( − 96 . 5 32 . 4 2 ) 2 ∗ 0 . 05 2 = 0 . 01 g/cm 3 (18) Density with triple beam balance and metric ruler: ρ = 2 . 98 ± 0 . 01 g/cm 3 (19) Uncertainty of density for digital balance and vernier calipers: Density = 96 . 71 37 . 01 = 2 . 61 g/cm 3 (20) σ ρ = r ( 1 v ) 2 ∗ 0 . 001 2 + ( − m v 2 ) 2 ∗ 0 . 001 2 (21) 5

σ ρ = r ( 1 37 . 01 ) 2 ∗ 0 . 001 2 + ( − 96 . 71 37 . 01 2 ) 2 ∗ 0 . 001 2 = 2 . 79 × 10 4 g/cm 3 (22) Density with digital and vernier calipers: ρ = 2 . 61 ± 2 . 79 × 10 − 4 g/cm 3 (23) Uncertainty of density for digital balance and micrometer: Density = 96 . 71 35 . 81 = 2 . 7 g/cm 3 (24) σ ρ = r ( 1 v ) 2 ∗ 0 . 05 2 + ( − m v 2 ) 2 ∗ 0 . 05 2 (25) σ ρ = r ( 1 35 . 81 ) 2 ∗ 0 . 001 2 + ( − 96 . 71 35 . 81 2 ) 2 ∗ 0 . 001 2 = 2 . 8 × 10 4 g/cm 3 (26) Density with digital and vernier calipers: ρ = 2 . 7 ± 2 . 8 × 10 − 4 g/cm 3 (27) 7 Conclusion For the three densities, our group had the following numbers: Density of triple beam balance + metric ruler 2.98: ± 0.01 g/cm 3 Density of digital balance + vernier calipers: 2.61 ± 0.01 g/cm 3 Density of digital balance + micrometer: 2.7 ± 2 . 8 e − 4 g/cm 3 Percent errors: • For the density of triple beam balance and metric ruler there was a 10.41% error. • For the density of digital balance and vernier calipers there was a 3.3% error. 6

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• For the density of digital balance and micrometer there was a 0.04% error. In the end, we determined that the digital balance combined with the micrometer had the most accurate measurement of the density of the alu- minum block. The density of aluminum is 2.7 g/cm 3 and has a percent error of 0.04% which is very accurate. After the micrometer, the vernier calipers with the digital balance had the second best accuracy with a density of 2.61 g/cm 3 and percent error of 3.4%. The triple beam balance and metric ruler had the least accurate density of 2.98 and percent error (10.41%). Overall, we learned how accurate each tool was to the theoretical value and now under- stand how important the error propagation formula is to find uncertainties. I expected the accuracy of all three of the densities to be like that because of the magnitude of each measuring devices’ uncertainties. The bigger the uncertainty of a measuring device, the less accurate the density would be. For some experimental errors, I think we had a couple. For the metric ruler, I think we didn’t gauge the right millimeter mea- surements as it is very hard to do with the naked eye. For the triple beam balance, the pointer might have not been matched with the zero line so there is a bit of error. These errors caused our density numbers to be significantly off for the metric ruler and triple beam balance density. Our result was big- ger than the theoretical value, but we expected this to happen. It is common to be off when your tool is not the most accurate. 7