Provided Data for Experiment 1 (Report_1_PHY2048L_data_summer_26)
Provided data for Exp 1 and instructions for data analysis and lab report. The raw measurements below are given in the PHY2048L data sheet for Summer 2026.
Table 1 — Steel ball (provided data)
| 1 | 2 | 3 | 4 | 5 | average |
|---|
| D (mm) | 15.885 | 15.880 | 15.880 | 15.880 | 15.880 | |
| V = πD³/6 | |
| |d_Vi| = |Vi − V̄| | |
| m (g) | 16.650 | 16.650 | 16.650 | 16.650 | 16.650 | |
| ρ (g/mm³) | |
| |d_ρi| = |ρi − ρ̄| | |
| % error between the measured ρ̄_Fe and the accepted ρ_Fe = 7.8 × 10³ kg/m³ : |
Table 2 — Aluminum block (provided data)
| 1 | 2 | 3 | 4 | 5 | average |
|---|
| L₁ (cm) | 1.550 | 1.550 | 1.550 | 1.550 | 1.550 | |
| L₂ (cm) | 1.560 | 1.550 | 1.540 | 1.550 | 1.540 | |
| L₃ (cm) | 4.880 | 4.890 | 4.880 | 4.870 | 4.880 | |
| V = L₁ · L₂ · L₃ | |
| |d_Vi| = |Vi − V̄| | |
| m (g) | 32.685 | 32.685 | 32.685 | 32.685 | 32.685 | |
| ρ (g/cm³) | |
| |d_ρi| = |ρi − ρ̄| | |
| % error between the measured ρ̄_Al and the accepted ρ_Al = 2.7 × 10³ kg/m³ : |
Table 3 — Brass cylinder (provided data)
| 1 | 2 | 3 | 4 | 5 | average |
|---|
| D (mm) | 12.670 | 12.665 | 12.670 | 12.660 | 12.660 | |
| L (cm) | 3.890 | 3.880 | 3.890 | 3.880 | 3.880 | |
| V = πD²L/4 | |
| |d_Vi| = |Vi − V̄| | |
| m (g) | 43.800 | 43.790 | 43.790 | 43.790 | 43.790 | |
| ρ (g/mm³) | |
| |d_ρi| = |ρi − ρ̄| | |
| % error between the measured ρ̄_Brass and the accepted ρ_Brass = 8.9 × 10³ kg/m³ : |
Table 4 — Aluminum annular cylinder (provided data)
| 1 | 2 | 3 | 4 | 5 | average |
|---|
| L (cm) | 0.940 | 0.940 | 0.940 | 0.940 | 0.940 | |
| D₁ (cm) | 1.880 | 1.890 | 1.870 | 1.880 | 1.870 | |
| D₂ (cm) | 0.960 | 0.970 | 0.970 | 0.970 | 0.960 | |
| V = (πL/4)(D₁² − D₂²) | |
| |d_Vi| = |Vi − V̄| | |
| m (g) | 5.720 | 5.720 | 5.720 | 5.720 | 5.720 | |
| ρ (g/cm³) | |
| |d_ρi| = |ρi − ρ̄| | |
| % error between the measured ρ̄_Al and the accepted ρ_Al = 2.7 × 10³ kg/m³ : |
Data analysis instructions for Experiment 1
(a) To reduce the measurement errors, five measurements have been made for each object as shown in each of the above 4 tables. Calculate the average (mean) dimensions of each object and record your calculated data in each Table.
(b) Calculate the volume of each object (V̄ ± d̄V), where V̄ is the mean of volume and d̄V is the mean deviation of volume. Record your calculated data in each Table.
(c) Calculate the density (ρ̄ ± d̄ρ) of the material of each object, where ρ̄ is the mean of density and d̄ρ is the mean deviation of density. Record your calculated data in each Table.
(d) Compare the measured ρ̄ with accepted ρ of each object and calculate the percent %. Record your calculated data in each Table.
(e) Put attention to the significant figures of your calculated data.
Lab report on Experiment 1
(a) Tables 1 to 4 with analyzed data must be included in your lab report.
(b) Answers to the questions #1 & #2 at the end of Exp 1 lab manual must be included in your lab report.
(c) The required other contents and format for your lab report can be found in the syllabus.
Experiment 1 — Completed Lab Report
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1. Introduction
The density of a body is calculated using the equation: ρ = m / V, where m is the mass of the object and V is the volume of the object.
To determine the volume, the object's dimensions (such as length, diameter, or radius) are first measured using instruments like the vernier caliper and micrometer screw gauge. These measured dimensions are then substituted into the appropriate geometric formula to calculate the volume. The mass of the object is measured using a laboratory balance.
The total reading from a vernier caliper is obtained using:
Total reading = Main scale reading + (Least count) × (smaller marking # on vernier scale that lines up or almost lines up with some main scale marking)
Where the vernier scale division is the number of the vernier marking that exactly (or nearly) aligns with a marking on the main scale. The least count is the smallest measurable value of the instrument.
The total reading from a micrometer is calculated as:
Total reading = Main scale reading + (Least count) × (Marking # on circular scale that is just below the central line on main scale)
Where the circular scale division is the marking on the circular scale that lies just below the central reference line on the main scale. The least count represents the smallest marked division measurable by the instrument. The least count of an instrument is defined as the smallest measurement that can be read directly from that instrument.
If the experimentally measured average value of a quantity is x̄, and the accepted (true) value is A, then the percentage error is calculated as:
% error = (|x̄ − A| / A) × 100
This value indicates how close the experimental result is to the accepted standard value.
2. Hypothesis
If the mass of an object which is measured using a laboratory balance is divided by its volume, which is calculated using dimensions such as length, diameter, etc. measured using vernier calipers and micrometer, then the quotient gives the density of the object.
Post-lab section
Table 1 — Steel ball (completed)
| 1 | 2 | 3 | 4 | 5 | average |
|---|
| D (mm) | 15.885 | 15.880 | 15.880 | 15.880 | 15.880 | 15.881 |
| V = πD³/6 (mm³) | 2098.75 | 2096.76 | 2096.76 | 2096.76 | 2096.76 | 2097.16 |
| |d_Vi| (mm³) | 1.59 | 0.40 | 0.40 | 0.40 | 0.40 | 0.638 |
| m (g) | 16.650 | 16.650 | 16.650 | 16.650 | 16.650 | 16.650 |
| ρ (g/mm³) | 0.00793 | 0.00794 | 0.00794 | 0.00794 | 0.00794 | 0.00794 |
| |d_ρi| (g/mm³) | 0.00001 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| % error between the measured ρ̄_Fe and the accepted ρ_Fe = 7.8 × 10³ kg/m³ : 1.79% |
Table 2 — Aluminum block (completed)
| 1 | 2 | 3 | 4 | 5 | average |
|---|
| L₁ (cm) | 1.550 | 1.550 | 1.550 | 1.550 | 1.550 | 1.550 |
| L₂ (cm) | 1.560 | 1.550 | 1.540 | 1.550 | 1.540 | 1.548 |
| L₃ (cm) | 4.880 | 4.890 | 4.880 | 4.870 | 4.880 | 4.880 |
| V = L₁·L₂·L₃ (cm³) | 11.800 | 11.748 | 11.649 | 11.700 | 11.649 | 11.709 |
| |d_Vi| (cm³) | 0.091 | 0.039 | 0.060 | 0.009 | 0.060 | 0.052 |
| m (g) | 32.685 | 32.685 | 32.685 | 32.685 | 32.685 | 32.685 |
| ρ (g/cm³) | 2.770 | 2.782 | 2.806 | 2.794 | 2.806 | 2.792 |
| |d_ρi| (g/cm³) | 0.022 | 0.028 | 0.014 | 0.002 | 0.014 | 0.016 |
| % error between the measured ρ̄_Al and the accepted ρ_Al = 2.7 × 10³ kg/m³ : 3.41% |
Table 3 — Brass cylinder (completed)
| 1 | 2 | 3 | 4 | 5 | average |
|---|
| D (mm) | 12.670 | 12.665 | 12.670 | 12.660 | 12.660 | 12.665 |
| L (cm) | 3.890 | 3.880 | 3.890 | 3.880 | 3.880 | 3.884 |
| V = πD²L/4 (mm³) | 4909.48 | 4888.01 | 4904.48 | 4884.15 | 4884.15 | 4894.05 |
| |d_Vi| (mm³) | 15.43 | 6.04 | 10.43 | 9.90 | 9.90 | 10.34 |
| m (g) | 43.800 | 43.790 | 43.790 | 43.790 | 43.790 | 43.792 |
| ρ (g/mm³) | 0.00892 | 0.00896 | 0.00893 | 0.00897 | 0.00897 | 0.00895 |
| |d_ρi| (g/mm³) | 0.00003 | 0.00001 | 0.00002 | 0.00002 | 0.00002 | 0.00002 |
| % error between the measured ρ̄_Brass and the accepted ρ_Brass = 8.9 × 10³ kg/m³ : 0.56% |
Table 4 — Aluminum annular cylinder (completed)
| 1 | 2 | 3 | 4 | 5 | average |
|---|
| L (cm) | 0.940 | 0.940 | 0.940 | 0.940 | 0.940 | 0.940 |
| D₁ (cm) | 1.880 | 1.890 | 1.870 | 1.880 | 1.870 | 1.878 |
| D₂ (cm) | 0.960 | 0.970 | 0.970 | 0.970 | 0.960 | 0.966 |
| V = (πL/4)(D₁²−D₂²) (cm³) | 1.929 | 1.943 | 1.887 | 1.915 | 1.901 | 1.915 |
| |d_Vi| (cm³) | 0.014 | 0.028 | 0.028 | 0.000 | 0.014 | 0.017 |
| m (g) | 5.720 | 5.720 | 5.720 | 5.720 | 5.720 | 5.720 |
| ρ (g/cm³) | 2.97 | 2.94 | 3.03 | 2.99 | 3.01 | 2.99 |
| |d_ρi| (g/cm³) | 0.02 | 0.05 | 0.04 | 0.00 | 0.02 | 0.03 |
| % error between the measured ρ̄_Al and the accepted ρ_Al = 2.7 × 10³ kg/m³ : 10.7% |
Sample calculations
From Table 1: Steel ball
ρ̄ (g/mm³) = m̄ / V̄ = 16.650 g / 2097.16 mm³ = 0.00794 g/mm³
ρ̄ (kg/m³) = 7.94 × 10³ kg/m³
Percentage error = |7.94 × 10³ − 7.8 × 10³| / (7.8 × 10³) × 100 = 1.79%
From Table 2: Aluminum block
ρ̄ (g/cm³) = m̄ / V̄ = 32.685 g / 11.709 cm³ = 2.792 g/cm³
ρ̄ (kg/m³) = 2.792 × 10³ kg/m³
Percentage error = |2.792 × 10³ − 2.7 × 10³| / (2.7 × 10³) × 100 = 3.41%
From Table 3: Brass cylinder
ρ̄ (g/mm³) = m̄ / V̄ = 43.792 g / 4894.05 mm³ = 0.00895 g/mm³
ρ̄ (kg/m³) = 8.95 × 10³ kg/m³
Percentage error = |8.95 × 10³ − 8.9 × 10³| / (8.9 × 10³) × 100 = 0.56%
From Table 4: Aluminum annular cylinder
ρ̄ (g/cm³) = m̄ / V̄ = 5.720 g / 1.915 cm³ = 2.99 g/cm³
ρ̄ (kg/m³) = 2.99 × 10³ kg/m³
Percentage error = |2.99 × 10³ − 2.7 × 10³| / (2.7 × 10³) × 100 = 10.7%
6. What I learned / Conclusion
From this laboratory experiment, I learned how the density of a material can be determined experimentally by accurately measuring its mass and volume and applying the relationship ρ = m / V. The use of precision instruments such as the vernier caliper, micrometer screw gauge, and laboratory balance emphasized the importance of measurement accuracy and the impact of uncertainty on experimental results.
The calculated densities for steel, aluminum, brass, and the aluminum annular cylinder were generally close to their accepted standard values. The percentage errors ranged from very small (such as 0.56% for brass) to higher values (such as 10.7% for the annular aluminum cylinder). These variations demonstrate how irregular shapes, small dimensions, and measurement uncertainties can significantly influence the calculated volume and, consequently, the density. However, all errors remained within the allowable experimental range.
7. Comparison with hypothesis
The experimental results support the hypothesis. Dividing the measured mass of each object by its calculated volume consistently produced density values that closely matched the accepted densities of the respective materials, with any deviations reasonably attributed to measurement uncertainty.
Questions from the Lab Manual
Q.1
a) Meter stick (cm): Reading = 15 cm. The least count is 1 cm.
b) Vernier Caliper (cm): Reading = 1.0 + (5 × 0.01) = 1.05 cm. The least count is 0.01 cm.
c) Micrometer gauge (mm): Reading = 5.5 + (28 × 0.01) = 5.78 mm. The least count is 0.01 mm.
Q.2
There is a difference in precision between the measured values 1.05 m and 1.050 m. The measured value of 1.05 m has three significant figures, but the measured value of 1.050 m has four significant figures, and therefore the latter has greater precision. The least count of the measuring instrument determines the number of significant figures of the measured value.
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