# Elementary Measurements of Mass, Volume, and Density Using Three Metal Cylinders ## Introduction The experiment aimed to determine the mass, volume, and density of three metal cylinders: brass, copper, and steel. Standard laboratory instruments, such as a balance and a vernier caliper, were used to obtain measurements. The analysis included statistical treatment of data and a discussion of measurement uncertainties. --- ## Experimental Method ### 1. Measurement of Mass - Each cylinder was placed on a digital balance. - The mass was recorded in grams (g). - Multiple measurements (typically three) were taken for each cylinder to ensure reliability. ### 2. Measurement of Dimensions - The diameter and height of each cylinder were measured using a vernier caliper. - Measurements were taken at different points to account for possible irregularities. - The average value for each dimension was calculated. ### 3. Calculation of Volume The volume \( V \) of each cylinder was calculated using the formula: \[ V = \pi \left(\frac{d}{2}\right)^2 h \] where: - \( d \) = average diameter, - \( h \) = average height. ### 4. Calculation of Density The density \( \rho \) was determined using: \[ \rho = \frac{m}{V} \] where: - \( m \) = average mass, - \( V \) = calculated volume. --- ## Data Analysis ### 1. Statistical Treatment - For each set of measurements (mass, diameter, height), the mean (\( \bar{x} \)), standard deviation (\( s \)), and uncertainty (\( u \)) were calculated: - Mean: \( \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i \) - Standard deviation: \( s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2} \) - Uncertainty: \( u = \frac{s}{\sqrt{n}} \) - The uncertainty in volume was estimated using error propagation: \[ \frac{\Delta V}{V} = 2\frac{\Delta d}{d} + \frac{\Delta h}{h} \] where \( \Delta d \) and \( \Delta h \) are the uncertainties in diameter and height, respectively. - The uncertainty in density was calculated as: \[ \frac{\Delta \rho}{\rho} = \frac{\Delta m}{m} + \frac{\Delta V}{V} \] ### 2. Sample Results Table | Metal | Mean Mass (g) | Mean Diameter (cm) | Mean Height (cm) | Volume (\( \text{cm}^3 \)) | Density (\( \text{g/cm}^3 \)) | Uncertainty (\( \text{g/cm}^3 \)) | |--------|---------------|--------------------|------------------|----------------------------|-------------------------------|-----------------------------| | Brass | ... | ... | ... | ... | ... | ... | | Copper | ... | ... | ... | ... | ... | ... | | Steel | ... | ... | ... | ... | ... | ... | *Note: Replace "..." with actual calculated values from experimental data.* --- ## Discussion ### 1. Sources of Inaccuracy - **Instrumental Error:** The precision of the digital balance and vernier caliper limits the accuracy of mass and dimension measurements. - **Human Error:** Parallax errors can occur during reading of the caliper scale. - **Cylinder Uniformity:** Slight imperfections or irregularities in shape can affect volume calculations. - **Temperature Variations:** Changes in temperature can affect both mass (due to air buoyancy) and dimensions (due to thermal expansion). ### 2. Statistical Treatment - Multiple measurements and calculation of standard deviation minimized random errors. - Error propagation formulas quantified how measurement uncertainties affected the final density values. ### 3. Reliability of Results - Comparison of the calculated densities with standard values for brass, copper, and steel indicated the accuracy of the measurements. - Larger deviations suggested the presence of systematic errors or imperfections in the experimental setup. --- ## Conclusion Elementary measurements of mass, volume, and density for the three metal cylinders were conducted systematically. Statistical analysis and error propagation were applied to quantify uncertainties. The main sources of inaccuracy included instrumental limitations and geometric irregularities of the samples. The experiment demonstrated fundamental measurement techniques in the determination of physical properties.