Chapter 7 Solved Numericals: Density and Temperature

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Chapter Overview & SLOs

What is covered in these solved numericals? This section provides comprehensive step-by-step solutions to the numerical problems for Chapter 7, "Density and Temperature." These exercises focus on the relationship between mass, volume, and density across various substances such as wood, metal, oil, plastic, and stone. How do we calculate density using the density formula? You will learn the fundamental density formula: - Formula: $ρ = \frac{m}{V}$ (density = mass / volume) - Rearranged formulas: $m = ρ × V$ (mass = density × volume), $V = \frac{m}{ρ}$ (volume = mass / density) - Unit conversions: 1 g/cm³ = 1000 kg/m³, 1 cm³ = 1 mL, 1 m³ = 1,000,000 cm³ - Convert cm³ to m³: divide by 1,000,000 (or multiply by 10⁻⁶) - Convert g/cm³ to kg/m³: multiply by 1000 How do we calculate volume of regular shapes? You will find step-by-step calculations for the volume of regular geometric shapes: - Cube: $V = a³$ (a = side length) - Rectangular box (cuboid): $V = l × b × h$ (length × breadth × height) - Sphere: $V = \frac{4}{3}πr³$ where radius $r = \frac{diameter}{2}$ - Cylinder: $V = πr²h$ (where r = radius, h = height) How do we use the displacement method for irregular objects? A key highlight includes using the displacement method with a measuring cylinder: - Method: Fill measuring cylinder with water, record initial volume (V₁). Submerge irregular object completely, record final volume (V₂). Volume of object = V₂ - V₁ - Key equivalence: 1 mL = 1 cm³ (milliliter equals cubic centimeter) - Density of irregular object: $ρ = \frac{m}{V₂ - V₁}$ - Example: Stone, marble, irregular metal piece How do we solve cavity volume problems? Advanced problems include finding the volume of a cavity within a solid object: - Concept: If an object has an internal cavity (air pocket/hollow space), its actual mass is less than the theoretical mass of a solid object of the same material. - Step 1: Calculate theoretical volume of the object from external dimensions - Step 2: Calculate mass if object were solid = density × theoretical volume - Step 3: Actual mass is given (lighter due to cavity) - Step 4: Mass of missing material = theoretical mass - actual mass - Step 5: Cavity volume = mass of missing material / density = (m_theoretical - m_actual) / ρ - Alternative method: Find actual volume from actual mass = m_actual / ρ. Cavity volume = theoretical volume - actual volume Why are unit conversions important? These numericals emphasize critical unit conversions: - Convert cm³ to m³: divide by 1,000,000 (or multiply by 10⁻⁶) - Convert g/cm³ to kg/m³: multiply by 1000 - Convert mL to cm³: 1 mL = 1 cm³ (direct equality) These solutions are strictly designed to help students master the mathematical requirements of the FBISE 2026 annual examination.

• How do we calculate the density of solids and liquids? Calculate the density of various solids and liquids using ρ = m/V given their mass and dimensions (for regular shapes) or volume (for liquids), with proper unit conversions including converting cm³ to m³ (divide by 1,000,000) and g/cm³ to kg/m³ (multiply by 1000).
• How do we apply geometric volume formulas to find mass? Apply geometric volume formulas for spheres (V = 4/3 πr³ where r = d/2) and cubes (V = a³) to determine the mass of objects based on their material density using m = ρ × V.
• How do we determine volume and density of irregular objects using displacement? Determine the volume and density of irregular objects (like stones) using the water displacement method in a measuring cylinder, where volume of object = final volume - initial volume (V₂ - V₁), and 1 mL = 1 cm³.
• How do we solve cavity volume problems in solid objects? Solve multi-step problems to find the volume of internal cavities in solid objects by comparing actual mass to theoretical mass using cavity volume = (m_theoretical - m_actual) / ρ, or cavity volume = theoretical volume - (m_actual / ρ).

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