Work, Energy & Power

Imagine this: You push a heavy almirah from one corner of room to other. You get tired, sweat. You did work. Now you carry a heavy bag on your head and walk 100m on flat road. You also get tired, but in physics, you did ZERO work. Confusing? That's because physics definition of work is very specific.

This whole chapter is about one simple idea: Energy is the capacity to do work. Work is transfer of energy. You eat paratha, you get chemical energy, you use it to push almirah, you transfer energy to almirah as work. That's it.

Let's break it like we are talking in chai tapri, not classroom.

Physics says work happens only when force AND displacement both happen, and force has a component along displacement.

Formula: W = F × s × cosθ where θ is angle between force and displacement.

Three cases you must feel:

Case 1 - θ=0° cos=1 Max positive work: You push box forward, box moves forward. You push in same direction as motion. Force helps motion. Like you pushing cycle from behind while friend rides. Work positive. You give energy to cycle.

Case 2 - θ=90° cos=0 Zero work: You carry bag on head and walk horizontal. Force you apply is upward (to hold bag against gravity), displacement is horizontal. 90° angle, cos90=0, so work ZERO. You feel tired because your muscles do internal work, but physics work on bag is zero. Same with coolie walking on platform with luggage on head - zero work. Also moon going around earth - gravity towards centre, motion tangential 90°, so gravity does zero work.

Case 3 - θ=180° cos=-1 Negative work: Force opposite to motion. Like friction when you slide box - friction backward, displacement forward, θ=180°, work negative. Friction takes away energy. Brakes on car do negative work.

Unit is Joule J = Newton × meter. 1 Joule is like lifting a small apple 1 meter up.

Work is scalar, no direction, but can be positive, negative, zero.

Constant force easy - same F whole time. But spring force changes as you stretch more. You pull rubber band - starting easy, later harder. Force not constant, variable.

How to calculate work then? Graph trick: Draw F vs x graph, area under curve = work. For constant force, rectangle area F×s. For variable, count area.

Spring is special: Hooke's law F = -k x. Minus means opposite to displacement. k is spring stiffness - hard spring high k. If you stretch spring by x, force by spring backward = k x, work done by YOU to stretch = +½ k x², work done BY spring = -½ k x².

Think of slingshot (gulel): You stretch rubber 10cm, you store energy ½ k x². Leave, rubber does work on stone and converts to kinetic.

Also work by gravity: Lifting mass m to height h against gravity g, work done by you = +m g h, work by gravity = -m g h negative because gravity down displacement up.

Work by friction always negative because friction opposite displacement, W = -f × s, energy lost as heat.

Kinetic Energy KE = ½ m v² - energy due to motion. Moving object has KE. Faster or heavier = more KE. Stationary object KE zero.

Why ½ m v²? Derive from work: you push mass m with force F over distance s, it accelerates from u to v. Using v²=u²+2 a s and F=m a, work W = F s = m a s = ½ m (v²-u²). So work done becomes change in ½ m v². That ½ m v² we call kinetic energy.

Now Work-Energy Theorem: Net work done on object = Change in KE. W_net = KE_final - KE_initial = ΔKE.

This is super powerful for numericals. No need to find acceleration, just energy.

Example 1: Ball dropped from height h. Gravity does work mgh, so mgh = ½ m v² final. So v = √(2 g h). You get speed without time.

Example 2: Car braking: Car has KE ½ m v², brakes apply friction force f over distance s to stop. Work by friction -f s = 0 - ½ m v² negative, so stopping distance s = v² / (2 μ g) where f=μ m g. Double speed = 4 times stopping distance! That's why speed thrills but kills, braking distance grows as v².

Example 3: You push box, positive work increases KE, it speeds up. Friction does negative work decreases KE, slows down.

Potential Energy PE is stored energy due to position or configuration. Like water stored on terrace has potential to do work when falls.

Two main types you need:

1) Gravitational PE = m g h - height h above reference (ground). Higher = more PE. When you lift book to shelf, you store mgh as PE. When it falls, PE converts to KE. That's why water from dam top generates electricity - mgh converts to KE to spin turbine.

2) Spring PE = ½ k x² - stretched or compressed spring stores energy. x is displacement from natural length. Slingshot, bow-arrow, trampoline all use this.

Now Conservative vs Non-conservative forces: Gravity and spring are conservative - work done doesn't depend on path, only start and end points, and you can get back energy fully. If you lift book up and bring down to same height, net work by gravity zero, energy returned. Like ideal bank loan - you get back.

Friction is non-conservative - work depends on path length, energy lost as heat sound never returns. If you slide box around and back to start, friction work not zero, energy lost. Like money spent on auto fare - gone.

Conservation of Mechanical Energy: If only conservative forces work (no friction), total mechanical energy ME = KE + PE = constant. PE decreases KE increases and vice versa, sum same. Roller coaster best example: Top hill PE max KE zero (slow), bottom valley PE min KE max (fastest), middle medium both. Total same if no friction. That's why roller coaster can go without motor after first lift.

If friction present, ME decreases, converts to heat. Total energy still conserved including heat, but ME not conserved.

Potential energy curve: Graph U vs x. Where U minimum = stable equilibrium like ball in bowl - small push returns. Where U maximum = unstable like ball on hill top - small push falls away. Where flat = neutral.

Work tells how much energy transferred, Power tells how fast.

Two people lift same bag to same height same work mgh, but one does in 5 sec, other in 20 sec. First is more powerful. So Power = Work / Time.

Average Power = Total Work / Total Time, Instantaneous Power = F · v = F v cosθ dot product. If you push with force F and move with velocity v, power = F v if same direction.

Unit Watt W = Joule/second. 1 kW = 1000 W. Old unit horsepower HP: 1 HP = 746 W. Your bike 10 HP means 7460 W power.

Example: You run up stairs vs walk up - same mgh work, but running needs more power because less time, you get tired faster. Electricity bill counts energy kWh = power × time. 100W bulb for 10 hours = 1 kWh = 1 unit.

Power also tells capacity: 100W bulb dim, 1000W heater hot because higher power converts more energy per second.

Collision is when two bodies come close and interact for short time with large force. Like billiard balls, car crash, hammer hitting nail.

Two things conserved (sometimes): Momentum always conserved if no external force (isolated). Energy may or may not.

Types:

1) Elastic collision: Both momentum AND kinetic energy conserved. Total KE same before and after. Ideal case like billiard balls, atomic collisions. Sound heat negligible. Example: Two identical balls, one moving hits stationary - first stops, second moves with same velocity. Energy transferred fully.

2) Inelastic collision: Momentum conserved but KE NOT conserved, some KE lost as heat, sound, deformation. Most real collisions. Car crash - cars crumple, KE lost.

3) Perfectly inelastic: After collision both stick together and move as one mass. Maximum KE loss but still momentum conserved. Like bullet hitting wooden block and embedding, or clay balls sticking.

In 1D elastic collision formulas: If m1 u1 + m2 u2 = m1 v1 + m2 v2 (momentum) and ½ m1 u1² + ½ m2 u2² = ½ m1 v1² + ½ m2 v2² (KE). Solving gives v1 = (m1-m2)u1+2 m2 u2 / (m1+m2), v2 = (m2-m1)u2+2 m1 u1/(m1+m2). If m1=m2 and one at rest, they exchange velocities.

Coefficient of restitution e = (velocity of separation)/(velocity of approach) = (v2'-v1')/(u1-u2). e=1 elastic, e=0 perfectly inelastic, 0

In 2D collisions, momentum conserved separately in x and y.

Daily: Cricket ball hitting bat - slightly inelastic, some energy lost as sound. Newton's cradle - almost elastic, momentum transfers through balls.