Work, Energy & Power: Definitions, Calculations & Conservation

Work: When Force Causes Displacement

Definition: Work is done when a force acts on an object and the object moves in the direction of the force.

• Formula: $W=F\cdot s\cdot \cos \theta$ where:
  • $W$ = Work done (Joules, J)
  • $F$ = Force applied (Newtons, N)
  • $s$ = Displacement (meters, m)
  • $\theta$ = Angle between force and displacement

Key Points:

• Work is a scalar quantity (only magnitude, no direction).
• If $\theta =0^{\circ }$, work is maximum ($W=F\cdot s$).
• If $\theta =90^{\circ }$, work is zero (force is perpendicular to displacement).

Example 1: Work Done by a Student A student lifts a 2 kg book 1.5 m vertically.

• Force = Weight = $mg=2\times 10=20\text{N}$
• Displacement = 1.5 m (upwards)
• Angle $\theta =0^{\circ }$ (force and displacement are in the same direction)
• Work done = $W=F\cdot s=20\times 1.5=30\text{J}$

Example 2: Work Done by Friction A box slides 5 m on a rough surface with a frictional force of 10 N opposing motion.

• Angle $\theta =180^{\circ }$ (force opposes displacement)
• Work done = $W=F\cdot s\cdot \cos 180^{\circ }=10\times 5\times (-1)=-50\text{J}$ (Negative work means energy is lost as heat.)

Energy: The Capacity to Do Work

Types of Energy:

1. Kinetic Energy (KE): Energy due to motion.

   • Formula: $KE=\frac{1}{2}m{v}^2$
   • Depends on mass and velocity squared.

2. Potential Energy (PE): Energy due to position or state.

   • Gravitational PE: $PE=mgh$ (Depends on mass, gravity, and height)
   • Elastic PE: $PE=\frac{1}{2}k{x}^2$ (Depends on spring constant and extension/compression)

Example 3: KE of a Moving Car A 1000 kg car moves at 20 m/s.

• $KE=\frac{1}{2}\times 1000\times (20{)}^2=200,000\text{J}=200\text{kJ}$

Example 4: PE of a Book on a Shelf A 0.5 kg book is placed 2 m above the ground.

• $PE=mgh=0.5\times 10\times 2=10\text{J}$

Work-Energy Theorem

• Work done on an object = Change in its kinetic energy. $W_{\text{net}}=\Delta KE=K{E}_{\text{final}}-K{E}_{\text{initial}}$

Example 5: Applying Work-Energy Theorem A ball of mass 0.2 kg is thrown upward with an initial velocity of 10 m/s. It reaches a maximum height of 5 m.

• Initial KE = $\frac{1}{2}\times 0.2\times (10{)}^2=10\text{J}$
• Final KE at max height = 0 (momentarily at rest)
• Work done by gravity = $\Delta KE=0-10=-10\text{J}$ (Negative because gravity opposes motion.)

Power: Rate of Doing Work

Definition: Power is the rate at which work is done or energy is transferred.

• Formula: $P=\frac{W}{t}$ or $P=F\cdot v$ where:
  • $P$ = Power (Watts, W)
  • $W$ = Work done (J)
  • $t$ = Time (s)
  • $v$ = Velocity (m/s)

Example 6: Power of an Electric Motor An electric motor lifts a 50 kg crate 10 m in 5 seconds.

• Work done = $mgh=50\times 10\times 10=5000\text{J}$
• Power = $\frac{5000}{5}=1000\text{W}=1\text{kW}$

Example 7: Power of a Running Athlete A 60 kg athlete runs 100 m in 10 s.

• Work done against gravity (assuming no vertical displacement) = 0
• But if we consider muscular power, we can estimate: $P=F\cdot v$ (Assuming average force $F=200\text{N}$ and speed $v=10\text{m/s}$) $P=200\times 10=2000\text{W}$

Conservation of Mechanical Energy

• Total mechanical energy (KE + PE) remains constant if only conservative forces (like gravity) act.
• Non-conservative forces (like friction) dissipate energy as heat.

Example 8: Pendulum Motion A 1 kg ball swings from a height of 2 m.

• At top: $PE=mgh=1\times 10\times 2=20\text{J}$, $KE=0$
• At bottom: $PE=0$, $KE=20\text{J}$
• If friction is negligible, total energy remains 20 J.

flowchart TD
    A["Highest Point\n(PE max, KE = 0)"] -->|"Swings down"| B["Lowest Point\n(PE min, KE max)"]
    B -->|"Swings up"| C["Highest Point\n(PE max, KE = 0)"]
    C --> A
    caption "Energy conversion in a pendulum (no friction)"

Comparison Table: Work, Energy, and Power

NEB-Style Questions (Practice)

Short Answer Questions

1. Define work. When is work said to be negative? (Answer: Work is force × displacement × cos θ. Negative when force opposes displacement.)

2. What is the difference between kinetic and potential energy? Give one example of each. (Answer: KE is energy of motion (e.g., moving car), PE is stored energy (e.g., stretched spring).)

3. Why does a moving object slow down when friction acts on it? (Answer: Friction does negative work, reducing KE until it stops.)

Numerical Problems

4. A 500 N force acts on a box at 30° to the horizontal. The box moves 4 m horizontally. Calculate the work done. (Answer: $W=500\times 4\times \cos 30°=1732\text{J}$)

5. A 2 kg ball is dropped from a height of 10 m. What is its speed just before hitting the ground? (Ignore air resistance.) (Answer: Use $PE=KE$: $mgh=\frac{1}{2}m{v}^2$ → $v=\sqrt{2gh}=14\text{m/s}$)

6. An electric bulb of 60 W is used for 5 hours. How much energy does it consume? (Answer: $E=Pt=60\times 5\times 3600=1,080,000\text{J}=1.08\times 10^6\text{J}$)

Conceptual Questions

7. Can work be done when displacement is zero? Explain. (Answer: No, because $W=F\cdot s\cdot \cos \theta$. If $s=0$, $W=0$.)

8. Why is kinetic energy proportional to the square of velocity? (Answer: $KE=\frac{1}{2}m{v}^2$. Doubling speed increases KE by 4 times.)

Exam Tip

✅ For NEB exams:

• Always draw diagrams for problems involving forces, displacements, or energy conversions.
• Label all given and asked quantities clearly in numerical problems.
• Remember units! Answers without units get zero marks.
• Conservation of energy is a high-scoring topic—always check if energy is conserved in a problem.
• Negative work is common in friction problems—don’t forget the cos θ term.

🔹 Common Mistakes to Avoid:

• Forgetting to convert units (e.g., kg to g, m/s to cm/s).
• Ignoring the angle $\theta$ in work calculations.
• Mixing up work and power (work is energy, power is energy per unit time).

Real-life examples like a roller coaster or bouncing ball. (Image: President (1977-1981 : Carter). White House Staff Photograph, Public domain, via Wikimedia Commons)