Lab Report No. 5

Camosun College Physics 104 Laboratory Report Objectives: 1. To determine the density of a steel ball bearing and a glass marble using a Vernier caliper, a micrometer, and a pan balance. 2. To calculate the absolute and relative uncertainties of the densities. 3. To compare the relative uncertainty in density of the ball bearing and the marble found with the two instruments. Data Tables and Calculations: 1. Table No. 1 – Ball Bearing Mass: 29.7 g ± 0.1 g VERNIER CALIPER (Diameter – cm) MICROMETER (Diameter – cm) 1.88 cm ±0.01 cm 1.883 cm ±0.001 cm 1.1 Calculation of the volume of the ball bearing and the uncertainty in the volume by each instrument used a. Vernier Caliper r = d 2 = 1.88 cm 2 ± 0.01 cm 2 = 0.94 ± 0.005 cm Relativeuncertainty = 0.005 cm 0.94 x 100% = 0.532% r = 0.94 cm± 0.532% V = 4 3 π r 3 →V = 4 3 π ( 0.94 cm± 0.532% ) 3 V = 4 3 π ( 0.94 cm ) ³ ± 3 ( 0.532% ) V = 4 3 π ( 0.831 cm 3 ) ± 1.596% V = 3.5 cm 3 ± 1.596% AbsoluteuncertaintyV = 0.1596% 100% x 3.5 cm 3 = 0.056 cm 3 1 Title: Errors and Precision Measurements Data Acquisition Date: October12th, 2023 Laboratory Report No.: 5 Author: Report Submission Date: October 26 th , 2023 Partners:

V = 3.5 ± 0.1 cm 3 b. Micrometer r = d 2 = 1.883 cm 2 ± 0.001 cm 2 = 0.9415 ± 0.0005 cm Relativeuncertainty = 0.0005 cm 0.9415 cm x 100% = 0.0531% r = 0.9415 cm± 0.0531% V = 4 3 π r 3 →V = 4 3 π ( 0.9415 cm± 0.0531% ) 3 V = 4 3 π ( 0.9415 cm ) ³ ± 3 ( 0.0531% ) V = 4 3 π ( 0.835 cm 3 ) ± 0.1593% V = 3.498 cm 3 ± 0.1593% Absoluteuncertainty V = 0.1593% 100% x 3.498 cm 3 = 0.0056 cm 3 V = 3.498 ± 0.0056 cm 3 →V = 3.5 ± 0.01 cm 3 1.2 Determination of the mass of the ball bearing and the uncertainty in the mass. m = 29.7 g± 0.1 g Relativeuncertaintym = 0.1 g 29.7 g x 100% = 0.337% m = 29.7 g± 0.337% 1.3 Calculation of the density of the ball bearing and the uncertainty in the density for each instrument. a. Vernier Caliper ρ = m v ρ = 29.7 g± 0.337% 3.5 cm 3 ± 1.596% → ρ = ( 29.7 g 3.5 cm 3 ) ± ( 0.337% + 1.596% ) 2

1.45 1.400 1.44 1.439 1.41 1.450 1.43 1.438 1.45 1.439 1.45 1.450 Mean Diameter ¿ 1 2 ( Max reading + Minreading ) ¿ 1 2 ( 1.45 cm + 1.41 cm ) = 1.43 cm Mean Diameter ¿ 1 2 ( Max reading + Minreading ) ¿ 1 2 ( 1.450 cm + 1.400 cm ) = 1.425 cm Fluctuation uncertainty ¿ 1 2 ( Max reading − Minreading ) ¿ 1 2 ( 1.45 cm − 1.41 cm ) = 0.02 cm Fluctuation uncertainty ¿ 1 2 ( Max reading − Minreading ) ¿ 1 2 ( 1.450 cm − 1.400 cm ) = 0.025 cm Total Uncertainty= Fluctuation uncertainty+ Measurement uncertainty = ±0.01cm+0.02cm = ±0.03cm Total Uncertainty= Fluctuation uncertainty+ Measurement uncertainty = ±0.001cm+0.025cm = ±0.026cm Volume %Diameter = 0.03 cm 1.43 cm x 100% = 2.0979% D = 1.43 cm ± 2.098% ¿ π 6 D 3 = π 6 ( 1.43 ± 2.098% ) 3 1.43 cm ¿ ¿ ¿ π 6 ¿ ¿ 1.53 cm 3 ± 6.294% ≈ 1.53 cm 3 ± 6.3% | uncer | .V = 6.3% 100% x 1.53 = 0.096 cm 3 ¿ 1.53 cm 3 ± 0.096 cm 3 ≈ 1.5 ± 0.1 cm 3 Volume %Diameter = 0.026 cm 1.425 cm x 100% = 1.8246% D = 1.425 cm ± 1.825% ¿ π 6 D 3 = π 6 ( 1.425 ± 1.825% ) 3 1.425 cm ¿ ¿ ¿ π 6 ¿ ¿ 1.52 cm 3 ± 5.475% ≈ 1.52 cm 3 ± 5.5% | uncer | .V = 5.5% 100% x 1.52 = 0.084 cm 3 ¿ 1.52 cm 3 ± 0.084 cm 3 ≈ 1.5 ± 0.1 cm 3 Mass m = 4.0 g± 0.1 g Mass m = 4.0 g± 0.1 g 4

Reluncer .m = 0.1 g 4.0 g x 100% = 2.5% m = 4.0 g± 2.5% Reluncer .m = 0.1 g 4.0 g x 100% = 2.5% m = 4.0 g± 2.5% Density ρ = m v ρ = 4.0 g± 2.5% 1.53 cm 3 ± 6.3% ρ = 4.0 g 1.53 cm 3 ± ( 2.5% + 6.3% ) ρ = 2.6 g cm 3 ± 8.8% | uncer | .ρ = 8.8% 100% x 2.6 = 0.228 g / cm 3 ρ = 2.6 g cm 3 ± 0.2 g / cm 3 Density ρ = m v ρ = 4.0 g± 2.5% 1.52 cm 3 ± 5.5% ρ = 4.0 g 1.52 cm 3 ± ( 2.5% + 5.5% ) ρ = 2.6 g cm 3 ± 8.0% | uncer | .ρ = 8.0% 100% x 2.6 = 0.208 g / cm 3 ρ = 2.6 g cm 3 ± 0.2 g / cm 3 Comparison of each of the calculated densities using a consistency check. | A − B | ≤∆ A + ∆B | A − B | = | 2.6 g / cm 3 − 2.6 g / cm 3 | = 0 g / cm 3 ∆ A + ∆B = 0.2 g / cm 3 + 0.2 g / cm 3 = 0.4 g / cm 3 | A − B | ≤∆ A + ∆B→ 0 g / cm 3 ≤ 0.4 g / cm 3 The measurements are CONSISTENTS since the inequality is valid Discussion: The results obtained in the laboratory verify the theory of the experiment within an acceptable uncertainty. Based on the data obtained, the results confirm the theory of the experiment. Consistency in measurements and calculations is a clear indicator of the accuracy and reliability of the techniques and 5