Density

• The relationship between density, mass, and volume is expressed by the formula: $\rho = \frac{m}{V}$ where $\rho$ (rho) is density, $m$ is mass, and $V$ is volume.

• Standard Units: In the SI system, density is measured in kilograms per cubic meter ($kg/m^3$), though grams per cubic centimeter ($g/cm^3$) is also common in laboratory settings.

• Particle Theory: Density is determined by the mass of individual atoms/molecules and how closely they are packed. Solids are generally the most dense because particles are tightly bound, while gases are the least dense because particles are far apart.

• Temperature Effects: For most substances, increasing temperature causes expansion (increased volume), which leads to a decrease in density.

1. Regularly Shaped Solids

• Mass: Measure using a digital balance.
• Volume: Calculate using geometric formulas (e.g., $V = l \times w \times h$ for a cuboid, $V = \pi r^2 h$ for a cylinder).
• Calculation: Divide the measured mass by the calculated volume.

2. Irregularly Shaped Solids (Displacement Method)

• Mass: Measure using a digital balance before the object gets wet.
• Volume: Submerge the object in a displacement (Eureka) can filled to the spout. The volume of the displaced water collected in a measuring cylinder equals the volume of the object.
• Calculation: Ensure the measuring cylinder is read at eye level to avoid parallax error.

3. Liquids

• Mass: Measure the mass of an empty measuring cylinder, add the liquid, and measure the total mass. Subtract the empty cylinder mass to find the liquid's mass.
• Volume: Read the volume directly from the scale on the measuring cylinder.
• Calculation: Apply the density formula using the net mass and the observed volume.

• Density vs. Weight: Weight is a force dependent on gravity ($W = mg$), whereas density is a material property that remains constant regardless of the gravitational field.

• Relative Density (Specific Gravity): This is the ratio of a substance's density to the density of water. It is a dimensionless number used to determine if a substance will float (ratio < 1) or sink (ratio > 1) in water.

• Unit Consistency: Always check if mass is in $kg$ and volume is in $m^3$ before calculating. If the question provides $g$ and $cm^3$, you may need to convert to $kg/m^3$ by multiplying the result by $1000$.

• Formula Rearrangement: Use the formula triangle to solve for different variables: $m = \rho \times V$ and $V = \frac{m}{\rho}$.

• Sanity Checks: Remember that the density of water is approximately $1000 kg/m^3$ (or $1 g/cm^3$). If your calculated density for a metal is less than this, re-check your volume calculations.

• Significant Figures: In practical exams, ensure your final answer matches the precision of your least precise measurement (usually 2 or 3 significant figures).

• Volume Conversion Error: A common mistake is assuming $1 m^3 = 100 cm^3$. In reality, $1 m^3 = (100 cm)^3 = 1,000,000 cm^3$.

• Zeroing the Balance: Forgetting to 'tare' or zero the digital balance when measuring liquids leads to systematic errors where the container's mass is included in the density calculation.

• Air Bubbles: When using the displacement method, air bubbles trapped on the surface of an irregular object will increase the apparent volume, leading to an incorrectly low density value.