Chapter 6 Solved Numericals: Work and Energy

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Chapter Overview & SLOs

What is covered in these solved numericals? This section provides comprehensive step-by-step solutions for the numerical problems of Chapter 6, "Work and Energy." The exercises guide students through fundamental calculations of work, energy transformations, power, and efficiency. How do we calculate work done and gravitational potential energy? You will learn fundamental calculations of: - Work done: $W = Fs$ (Work = Force × displacement in direction of force) - Unit conversions: Convert cm to m (divide by 100), g to kg (divide by 1000) - Gravitational potential energy: $PE = mgh$ (using $g = 9.8 \text{ m/s}^2$) - Finding height: $h = \frac{PE}{mg}$ How do we solve potential to kinetic energy transformation problems? You will find detailed applications of energy transformations: - Energy conservation: Initial PE = Final KE (when no energy losses) - Finding final velocity: $mgh = \frac{1}{2}mv^2$ → $v = \sqrt{2gh}$ - Note: Mass cancels out, so velocity depends only on height, not mass - Convert grams to kilograms: For example, a 800g rocket = 0.8 kg How do we apply the law of conservation of energy? A major focus is placed on the Law of Conservation of Mechanical Energy: - Total energy at various heights: At any point, $E_{total} = KE + PE = \text{constant}$ - Work done against resistive forces (air friction, friction): - Energy loss = Initial total energy - Final total energy - Work done against resistance = Energy loss - $W_{\text{resistance}} = \text{Initial PE} - \text{Final KE}$ (for falling objects) How do we calculate efficiency and power? The notes cover practical applications of Power and Efficiency: - Power formula: $P = \frac{W}{t}$ (Power = Work / time) in Watts (W = J/s) - Horsepower conversion: $1 \text{ hp} = 746 \text{ W}$ (approximately 745.7 W) - Efficiency formula: $\text{Efficiency} = \frac{\text{Useful Output Energy}}{\text{Total Input Energy}} × 100\% = \frac{\text{Useful Output Power}}{\text{Total Input Power}} × 100\%$ - Determine input and output work for electric motors, pumps, and other machines - Calculate power requirements for industrial pumps in horsepower These solutions emphasize the use of $g = 9.8 \text{ m/s}^2$ in all energy equations, proper unit conversions (converting grams to kilograms, cm to m), and the correct application of the conservation of energy principle. They are strictly designed to help students master the mathematical requirements of the FBISE 2026 annual examination.

• How do we calculate work done and gravitational potential energy? Calculate the work done by a force acting over a distance using W = Fs (with proper unit conversions: convert cm to m, g to kg), and determine height based on gravitational potential energy using h = PE/mg with g = 9.8 m/s².
• How do we solve problems involving potential to kinetic energy transformation? Solve problems involving the transformation of potential energy into kinetic energy using the law of conservation of energy (initial PE = final KE when no energy loss), and calculate the resulting impact velocity using v = √(2gh), noting that velocity is independent of mass.
• How do we apply the law of conservation of mechanical energy and calculate energy loss? Apply the law of conservation of mechanical energy to determine total energy (KE + PE = constant) at various heights, and calculate energy loss due to air resistance or friction using energy loss = Initial PE - Final KE (work done against resistive forces).
• How do we determine efficiency and power requirements? Determine the efficiency of mechanical systems using Efficiency = (Useful Output / Total Input) × 100%, and calculate power requirements in both Watts (P = W/t) and Horsepower using the conversion factor 1 hp = 746 W.

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