Density of solids lab report

1. Title and aim

Measurement of mass, volume and determination of the density of solids. Determine the density of pure copper specimens by linear regression of mass against volume using precision balance and displacement measurements.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of solid density of 8960.0 kg m⁻³, differing by 0.000% from the model value. The reported scatter comes from scale rounding and any stated model approximation, not measured laboratory noise.

3. Apparatus and fixed settings

• Precision laboratory balance: Measures specimen gravitational mass M with a resolution of 0.01 g.
• Graduated cylinder and Eureka displacement can: Measures volume of regular or irregular solids by water displacement with anti-meniscus correction.
• Vernier calipers: Measures diameter and height of cylindrical specimens to calculate volume independently.
• Set of calibrated copper specimens: Provides homogeneous test cylinders of graded volumes from 5.0 cm³ to 60.0 cm³.
• Specimen material: Standard electrolytic tough-pitch copper (reference density ρ = 8960 kg m⁻³ at 20 °C).
• Room temperature: 20.0 °C; atmospheric pressure: 101.3 kPa.
• Balance: Triple-beam balance with vernier rider (resolution 0.01 g).
• Volume determination: Precision graduated cylinder displacement and vernier caliper dimensional cross-check.

4. Theory and graph choice

The density of a homogeneous solid material is defined as its mass per unit volume: ρ = M/V. For a set of specimens of varying dimensions or immersed liquid volumes, the relationship between mass M and volume V is strictly linear with slope equal to the bulk density ρ and zero theoretical intercept.

The calculation fits y = mx + b in SI units. The graph may display scaled axes for readability; those display factors are removed before calculating the physical result. The gradient has units kg m⁻³.

5. Method

1. Release the balance beam arrest and adjust the counterpoise thumb screw until the pointer oscillates symmetrically about the central null mark.
2. Measure the height and diameter of each cylindrical specimen at three distinct orientations using a vernier caliper to verify geometric regularity.
3. Fill a graduated cylinder with distilled water to a reference level, tilt slightly to avoid air entrapment, lower the dry specimen gently on a fine thread, and record the new meniscus level.
4. Remove the specimen from the water bath and blot it dry with lint-free absorbent paper before placing it on the balance pan.
5. Place the dry specimen at the center of the pan, slide the poise weights along the beams until equilibrium is restored, and read the total mass to the nearest 0.01 g.
6. Proceed systematically through the graded series of specimen volumes, logging volume V and mass M for linear regression analysis.

6. Observations and calculation table

All observations use fixed model settings. Values are rounded only at the instrument reading stage; further calculations use full precision. Repeated physical trials are not claimed.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Dots show the sample observations. The line is the ordinary least-squares fit.

9. Uncertainty and comparison

Percentage difference = |estimate − model value| / |model value| × 100 = 0.0000%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 2.5037e-33; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 9.1355e-13. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 3.0299e-17.

Propagating only the relevant fit coefficient gives a fit-only standard error of 9.1355e-13 kg m⁻³. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Volumetric measurement by fluid displacement is subject to meniscus reading uncertainty, liquid adherence to cylinder walls, and thermal expansion. Mass measurements on a mechanical balance involve knife-edge friction and buoyancy in ambient air. The regression slope error accounts for linear scatter, but does not capture shared calibration offsets in the volumetric scale or balance tare.

10. Result and discussion

The observations follow the stated model relationship. Check whether the intercept and residual pattern suggest a systematic offset or a missing approximation. Agreement with the model does not independently verify its input constants.

11. Precautions and limitations

• Zero the balance carefully and ensure the knife edges and pans are free from dust before each weighing.
• Read the liquid meniscus in the graduated cylinder horizontally at the level of the lowest point of the concave water surface to avoid parallax.
• Ensure no air bubbles cling to the surface of the specimen when immersed in the displacement cylinder; dislodge any bubbles gently with a clean glass rod.
• Thoroughly dry each specimen with absorbent lint-free paper before placing it on the balance pan to avoid spurious water mass.

The simulation assumes an ideal homogeneous copper specimen without internal casting porosity, oxide films, or surface contamination. Readings in this virtual module reflect ideal copper (ρ = 8960 kg m⁻³) with standard rounding to 0.01 g on balance mass and 0.1 cm³ on volume.

12. Conclusion

The worked example illustrates how to obtain solid density from measurements, graph variables and a fitted coefficient. The estimate is 8960.0 kg m⁻³. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.