Work and Energy — SOF NSO Study Notes

Overview

Work and Energy is a foundational mechanics topic that accounts for 5–7% of SOF NSO questions, appearing in both the Science section (direct application) and Achievers section (multi-concept problems). This chapter bridges kinematics and dynamics by introducing the energy perspective—a powerful alternative to Newton's laws for solving motion problems.

Mastery requires three skills: (1) calculating work done by forces, (2) applying kinetic and potential energy formulas correctly, and (3) using conservation of energy to solve problems involving energy transformations. Questions test numerical computation, conceptual understanding of energy transformations, and identification of conservative vs non-conservative forces. Students often face scenarios involving falling objects, sliding bodies, springs, and pendulums where energy methods simplify calculations that would otherwise require complex force analysis.

The topic directly connects to real-world applications—hydroelectric power, roller coasters, and efficiency of machines—making it a favorite for HOTS questions in the Achievers section.

Key Concepts

• **Work** is done when a force causes displacement in its direction. If force and displacement are perpendicular (like centripetal force in circular motion), work done is zero. Work is a scalar quantity measured in joules.

• **Kinetic Energy (KE)** is the energy possessed by a body due to its motion. It depends on mass and the square of velocity, so doubling velocity quadruples kinetic energy. A moving bullet, flowing river, and spinning wheel all have kinetic energy.

• **Potential Energy (PE)** is the energy stored in a body due to its position or configuration. Gravitational PE increases with height above ground; elastic PE stores when you compress a spring or stretch a rubber band.

• **Conservation of Energy** states that energy can neither be created nor destroyed, only transformed from one form to another. The total mechanical energy (KE + PE) of a system remains constant in the absence of friction or air resistance.

• **Power** measures the rate of doing work or the rate of energy transfer. A more powerful engine does the same work in less time. Power determines how quickly energy is converted from one form to another.

• **Energy transformations** occur continuously—chemical energy in food converts to kinetic energy when you run, electrical energy becomes light and heat in a bulb, and potential energy of water at a height becomes kinetic energy in turbines.

Formulas / Key Facts

**Work (W)** = Force × Displacement × cos θ Where θ is the angle between force and displacement. If θ = 0° (same direction), W = F × s. If θ = 90°, W = 0.

**Kinetic Energy (KE)** = ½ mv² Where m is mass (kg) and v is velocity (m/s). Unit: joule (J).

**Gravitational Potential Energy (PE)** = mgh Where m is mass (kg), g is acceleration due to gravity (10 m/s²), h is height (m). Unit: joule (J).

**Work-Energy Theorem**: Work done by all forces = Change in kinetic energy W = KE_final − KE_initial = ½ m(v² − u²)

**Conservation of Mechanical Energy**: KE₁ + PE₁ = KE₂ + PE₂ Valid only when no non-conservative forces (friction, air resistance) act.

**Power (P)** = Work/Time = Energy/Time Unit: watt (W) = joule/second. Also, P = Force × velocity for constant force.

**1 kilowatt-hour (kWh)** = 3.6 × 10⁶ J Commercial unit of electrical energy used in electricity bills.

Worked Examples

**Example 1: Work Done by Gravity** A 5 kg stone falls from a height of 20 m. Calculate the work done by gravity.

*Solution:* Work = Force × displacement = (mg) × h = 5 × 10 × 20 = 1000 J Note: Gravity and displacement are in the same direction (downward), so cos θ = 1.

**Example 2: Kinetic and Potential Energy** A ball of mass 0.2 kg is thrown upward with velocity 20 m/s. Find (a) initial KE, (b) PE at maximum height, (c) maximum height reached.

*Solution:* (a) Initial KE = ½ mv² = ½ × 0.2 × (20)² = 40 J (b) At maximum height, velocity = 0, so KE = 0. By conservation of energy, all KE converts to PE. Therefore, PE at maximum height = 40 J (c) PE = mgh → 40 = 0.2 × 10 × h → h = 40/2 = 20 m

**Example 3: Power Calculation** A pump lifts 200 kg of water to a height of 5 m in 10 seconds. Calculate the power of the pump.

*Solution:* Work done = mgh = 200 × 10 × 5 = 10,000 J Power = Work/Time = 10,000/10 = 1000 W = 1 kW

Common Mistakes

**Mistake**: Calculating work as F × s when force and displacement are not parallel → **Fix**: Always use W = F × s × cos θ. For perpendicular force (like tension in circular motion at lowest point contributing no work in horizontal direction), work is zero.

**Mistake**: Thinking kinetic energy is directly proportional to velocity → **Fix**: KE is proportional to v². If velocity doubles, KE becomes four times, not two times. This is critical in collision and stopping-distance problems.

**Mistake**: Using conservation of energy when friction is present → **Fix**: Mechanical energy is conserved only in absence of friction. When friction acts, some mechanical energy converts to heat: Initial ME = Final ME + Energy lost to friction.

**Mistake**: Confusing mass with weight in PE formula, writing PE = wgh → **Fix**: PE = mgh, where m is mass in kg. Weight (w = mg) already includes g, so PE = wh would double-count gravity.

**Mistake**: Mixing units—using grams instead of kg or cm instead of m → **Fix**: Always convert to SI units before calculation: mass in kg, distance in m, force in N. Final answer for energy/work will be in joules.

Quick Reference

• Work is done only when force causes displacement in its direction; work is zero if force ⊥ displacement.
• KE = ½ mv² depends on the square of velocity; PE = mgh depends linearly on height.
• Total mechanical energy (KE + PE) is conserved when only conservative forces (gravity, spring force) act.
• Work-energy theorem: Net work done = Change in kinetic energy.
• Power = Work/Time; measured in watts; 1 kW = 1000 W; 1 kWh = 3.6 million joules.
• Energy cannot be created or destroyed—it only transforms (chemical → kinetic → heat, etc.).