Work and Energy – Study module with Revision Notes
- Work (W) = F · s · cosθ — Work done by a constant force F when displacement is s and θ is angle between them. Unit: joule (J).
- Kinetic energy (K) = ½ m v² — Energy due to motion.
- Potential energy near Earth (U) = m g h — Gravitational potential energy relative to a reference level.
- Work–energy theorem: Work_net = ΔK = K_f − K_i.
- Power (P) = W / t = F · v — Rate of doing work; unit watt (W) = J/s.
- Conservation of mechanical energy (in absence of non-conservative forces): K_i + U_i = K_f + U_f.
Comprehensive Revision Notes — Work and Energy
- What is Work? (Definition & examples)
- Kinetic Energy — derivation and examples
- Potential Energy — gravitational PE and reference levels
- Work–Energy Theorem
- Power — instantaneous & average
- Conservation of Mechanical Energy (applications)
- Common Numericals & Problem-solving strategy
- Exam tips & common pitfalls
1. What is Work?
In physics, work is said to be done when a force applied on a body produces displacement of the body in the direction of force. The scalar quantity work measures energy transfer by mechanical means.
Mathematical definition (constant force):
W = F s cos θ
where F is the magnitude of the applied force, s is the magnitude of the displacement and θ is the angle between the force vector and the displacement vector.
Important points:
- If θ = 0° (force and displacement same direction) → W = F s (positive work).
- If θ = 90° (force perpendicular to displacement) → W = 0. Example: normal force on a block sliding horizontally — does no work towards horizontal displacement.
- If θ = 180° (force opposite to displacement) → W = −F s (negative work) — force removes energy from the body (e.g., friction doing negative work on a moving block).
Examples of Work (qualitative)
- Pushing a box across the floor: applied horizontal force does positive work (if displacement in same direction).
- Holding a heavy bucket stationary: your supporting force does zero work because displacement = 0.
- Gravity does negative work on an object being lifted upward (opposite to displacement); it does positive work when object falls down.
2. Kinetic Energy (KE)
Kinetic energy is the energy possessed by a body due to its motion. For a body of mass m moving with speed v (non-relativistic), kinetic energy is:
K = ½ m v²
Derivation (brief): Use work–energy concept. Consider a constant net force accelerating mass from speed u to v over displacement s. From Newton: F = m a and kinematics v² − u² = 2 a s. Work done W = F s = m a s = ½ m (v² − u²). So the work done equals change in ½ m v², motivating definition of kinetic energy.
3. Potential Energy (PE)
Potential energy is energy stored in a body because of its position in a force field. In Class 9 NCERT context we focus on gravitational potential energy near Earth’s surface.
U = m g h
where h is height above a chosen reference level (often ground). Note that potential energy depends on the reference choice: only changes in potential energy (ΔU) are physically meaningful.
4. Work–Energy Theorem
The work–energy theorem states that the net work done by all forces acting on a body equals the change in its kinetic energy:
W_net = ΔK = ½ m v_f² − ½ m v_i²
This theorem is extremely useful in solving problems where forces cause acceleration: instead of using kinematics and dynamics separately, compute net work to get change in kinetic energy directly.
5. Power
Power is the rate at which work is done or energy is transferred. Average power over time t when work W is done:
P_avg = W / t
Instantaneous power delivered by a constant force F when the object moves with velocity v:
P = F · v
Unit: watt (W) where 1 W = 1 J/s. Common conversions: 1 horsepower ≈ 746 W.
6. Conservation of Mechanical Energy
When only conservative forces (like gravity, elastic spring force) do work, the total mechanical energy (kinetic + potential) of a system remains constant:
K_i + U_i = K_f + U_f
In the presence of non-conservative forces like friction, mechanical energy is not conserved; some mechanical energy is converted to internal energy (thermal).
7. How to solve typical numericals
Follow these steps:
- Read carefully: Identify what is given (m, v, h, F, s, θ) and what is asked.
- Choose method: Is it easier to use work = F s cosθ, work–energy theorem, or conservation of energy?
- Define reference level: For potential energy choose a convenient h = 0.
- Show steps: Write formulae, substitute values with units, calculate, and give final answer with proper units and reasonable significant figures.
Common NCERT-style numericals (short list)
- Compute work done by a constant force of 10 N moving object 3 m in same direction.
- Find speed of a mass m that has fallen from height h (use v = √(2gh)).
- Find power developed when a 500 N force moves object at speed 2 m/s.
- Using work–energy theorem: compute final speed given net work done and mass.
- Work W = m g h = 2 × 9.8 × 5 = 98 J. Increase in PE = 98 J. Since motion is at constant speed, net work = 0; lifting force did +98 J, gravity did −98 J.
8. Exam Tips & Common Pitfalls
- Define quantities clearly: Always state the reference level for potential energy when asked.
- Units matter: SI units — mass (kg), distance (m), time (s), work/energy (J), power (W).
- Signs: Use sign convention carefully — work done against gravity is positive for the agent doing the lift, gravitational work is negative when object is raised.
- Sketch diagrams: Free-body diagrams and energy-level sketches help in reasoning and scoring marks.
- Use work–energy theorem: It often shortens algebra for motion with variable forces or when acceleration is not given explicitly.
- Memorize key formulas: W = F s cosθ, K = ½ m v², U = m g h, P = W/t.
- Check extremes: For v → 0, KE → 0; for h → 0, PE → 0. This helps detect algebraic errors.