Work, Energy and Power

The chapter opens by fixing the everyday meaning of 'work' against the physics meaning. In physics, work is done only when a force produces a displacement in the direction of the force (or a component of it). A person holding a heavy suitcase and standing still does zero physics-work, however tired the muscles get, because there is no displacement. The chapter builds three ideas in sequence: work as a scalar defined by a dot product, energy as the capacity to do work (kinetic energy for motion, potential energy for position/configuration), and power as the rate of doing work. These three, along with the law of conservation of energy, are used throughout mechanics, so the definitions must be stated exactly as NCERT gives them, not loosely.

Work, energy and power are all scalar quantities even though force, displacement and velocity (which define them) are vectors. This scalar nature is what makes the work-energy theorem and conservation of energy such powerful problem-solving tools — you can add and subtract energies without worrying about direction.

NCERT introduces work-energy ideas informally first through three familiar situations (a moving cricket ball hitting a stationary wicket, a moving trolley pushed by a mass, a vehicle braking on the road) before formalising them in §5.3–5.4. Each example shows that the change in kinetic energy of a body equals the work done on it by the net force — this is the seed of the work-energy theorem proved rigorously later. The theorem is stated as: work done by the net force on a body equals the change in its kinetic energy, W = ΔKE = KE_f − KE_i. It holds for both constant and variable forces and is one of the most repeated results in the whole of mechanics.

These qualitative examples are important because they let you reason about direction of force versus motion without a formula: braking force acts opposite to velocity so it does negative work and reduces KE; a falling object has gravity doing positive work and gaining KE.

For a constant force F acting on a body while it undergoes a displacement d, work done is defined as the scalar (dot) product: W = F·d = F d cosθ, where θ is the angle between the force and the displacement. Because it is a dot product, work is a scalar — it has magnitude but no direction, even though F and d are vectors.

Three special cases matter for exams. When θ = 0° (force along displacement), W = Fd, maximum positive work. When θ = 90° (force perpendicular to displacement), W = 0 — this is why the work done by centripetal force in circular motion, by the normal reaction on a body moving along a surface, and by tension in a string whirling a stone, are all zero. When θ = 180° (force opposite to displacement), W = −Fd, negative work — as in friction opposing motion or gravity acting on a rising object.

For a variable force, work is computed by summing (integrating) F·dx over small displacements: W = ∫F dx from initial to final position. Graphically, this is the area under the F–x curve, with area below the x-axis counted as negative.

Kinetic energy is the energy possessed by a body by virtue of its motion. For a body of mass m moving with speed v, KE = ½mv². It is a scalar and is always positive (or zero), regardless of the direction of velocity — this is a favourite assertion-reason trap since momentum (mv) can be negative but KE (½mv²) cannot.

Using the equation of motion v² = u² + 2as for a body accelerated by a constant force F = ma from speed u to speed v over displacement s, work done by the force is W = Fs = mas = m(v² − u²)/2 s /s... more directly, W = ½mv² − ½mu² = ΔKE. This is the formal proof of the work-energy theorem for a constant force. NCERT extends this proof to a variable force using calculus (W = ∫F dx = ∫m(dv/dt)dx = ∫mv dv = ½mv² − ½mu²), which is the version most often asked as a 3–5 mark derivation.

Most real forces (a spring force, a varying push) change with position, so the simple W = Fd cosθ does not apply directly. NCERT's method: divide the total displacement into a large number of tiny intervals Δx_j, treat the force as constant (F_j) over each tiny interval, sum ΣF_jΔx_j, and take the limit as Δx → 0. This limit is the definite integral W = ∫_xi^x_f F(x) dx, and geometrically it is the area under the force–displacement curve between the initial and final positions.

This is the general definition of work; the constant-force formula and the spring's PE formula (§5.9) are both special cases of it. In numericals, when F(x) is given as an algebraic expression (e.g. F = kx or F = a + bx), integrate directly; when F is given as a graph, count the enclosed area (using simple shapes — triangles, rectangles, trapeziums).

This is the rigorous, most-asked derivation of the chapter. Start from Newton's second law F = m(dv/dt). Work done over a small displacement dx is dW = F dx = m(dv/dt) dx = m(dx/dt) dv = mv dv (since dx/dt = v). Integrating both sides from the initial speed u to the final speed v:

W = ∫_u^v mv dv = m[v²/2]_u^v = ½mv² − ½mu²

The right-hand side is exactly ΔKE, so W = ΔKE = KE_f − KE_i. This proves that the work-energy theorem, discovered for a constant force in §5.4, holds equally for a variable force — this generality is why the theorem is used so widely, including for springs and other non-uniform forces later in the chapter. Note that W here means the work done by the net (resultant) force on the body.

Potential energy is the energy stored in a body or system by virtue of its position or configuration, associated only with conservative forces (gravity, spring force, electrostatic force) — it cannot be defined for a non-conservative force like friction, because the work done by friction depends on the actual path taken, not just the end points. For a conservative force, the work done in moving a body from one point to another is independent of the path and depends only on the initial and final positions; the negative of this work is defined as the change in potential energy: W = −ΔPE i.e. ΔPE = −W.

The most familiar case is gravitational PE near Earth's surface: raising a body of mass m through height h against gravity, work done by the person = mgh, so PE gained = mgh (taking PE = 0 at the reference/ground level). PE is defined only up to an arbitrary additive constant — only differences (ΔPE) have physical meaning, which is why a reference level must always be chosen and stated.

For a body moving under a conservative force only (no friction, no air resistance), the total mechanical energy — kinetic plus potential — remains constant at every point of the motion: KE + PE = constant, i.e. KE_i + PE_i = KE_f + PE_f. The proof follows directly from combining the work-energy theorem (W = ΔKE) with the definition of PE (W = −ΔPE): equating the two gives ΔKE = −ΔPE, i.e. ΔKE + ΔPE = 0, so KE + PE stays constant.

The classic application is a body falling freely from height H. At the top, KE = 0, PE = mgH. At height h during the fall, KE = mg(H−h) [from v² = 2g(H−h)], PE = mgh, and their sum is always mgH. At the ground, all PE has converted to KE = mgH. This is tested heavily with a pendulum, a body sliding down a frictionless incline, or a block on a spring.

If a non-conservative force like friction also acts, mechanical energy is not conserved — the 'lost' mechanical energy appears as heat/sound (total energy, including these forms, is still conserved, which is the deeper law in §5.10).

A spring obeys Hooke's law: the restoring force is F = −kx, where x is the displacement of the free end from the natural (unstretched) length and k is the spring (force) constant (unit N m⁻¹), with the negative sign showing the force always opposes the displacement. The applied (external) force needed to stretch or compress it slowly is F_applied = +kx.

Work done in stretching the spring from x = 0 to x = x_m is W = ∫₀^x_m kx dx = ½kx_m². Since the spring force is conservative, this work is stored as elastic potential energy: PE(x) = ½kx². This PE is the same whether the spring is stretched or compressed by x (it depends on x², so it's always positive), and graphically PE(x) vs x is a parabola while F(x) vs x is a straight line through the origin with negative slope −k.

At maximum extension x_m, if the spring-mass system oscillates, KE = 0 and PE = ½kx_m² (all energy is potential); at x = 0 (natural length), PE = 0 and KE is maximum, equal to ½kx_m² by conservation.

Beyond mechanical energy, NCERT surveys other forms — heat energy (from molecular motion, measured via the mechanical equivalent of heat, 4.186 J = 1 calorie), chemical energy (stored in molecular bonds, released or absorbed in reactions, e.g. combustion), electrical energy, and the mass-energy equivalence E = mc² from special relativity (nuclear energy, where a small mass defect releases enormous energy, e.g. nuclear fission and fusion, and the Sun's energy source).

The law of conservation of energy is the deepest generalisation of this chapter: energy can only be transformed from one form to another; it can neither be created nor destroyed. The total energy of an isolated system remains constant. This is far more general than conservation of mechanical energy — it holds even when friction, heat and other dissipative or non-mechanical processes are present, because the 'lost' mechanical energy simply reappears as heat, sound or other forms, keeping the grand total unchanged.

Power is the time rate of doing work, i.e. the rate of transfer of energy. Average power over a time interval Δt is P_av = W/Δt. Instantaneous power is the limit as Δt → 0: P = dW/dt. Since dW = F·dr, instantaneous power can also be written as P = F·v, the dot product of the applied force and the instantaneous velocity — a very useful form for numericals involving vehicles, pumps and motors.

SI unit of power is the watt (W), 1 W = 1 J s⁻¹. A common commercial (non-SI) unit is the horsepower, 1 hp = 746 W. Dimensional formula of power is [M L² T⁻³]. The commercial unit of energy, the kilowatt-hour (kWh, colloquially a 'unit' of electricity), equals 3.6 × 10⁶ J.

A collision is an event where two (or more) bodies exert strong mutual forces for a short time, changing their velocities. Momentum is always conserved in every collision (no external force acts during the brief collision interval). Collisions are classified by whether kinetic energy is also conserved.

Elastic collision: both momentum and kinetic energy are conserved; bodies separate after collision without any permanent deformation or heat loss (an idealisation, e.g. collisions between atoms/billiard balls approximately). Inelastic collision: momentum is conserved but kinetic energy is not — some KE converts to heat, sound or deformation. In a perfectly inelastic collision, the colliding bodies stick together and move with a common velocity after impact, and the KE loss is maximum for the given momentum.

1-D elastic collision derivation (mass m₁ with initial velocity u₁ strikes stationary m₂, u₂=0): conservation of momentum gives m₁u₁ = m₁v₁ + m₂v₂; conservation of KE gives ½m₁u₁² = ½m₁v₁² + ½m₂v₂². Solving these simultaneously (rearranging each as a difference of squares and dividing) yields v₁ = [(m₁−m₂)/(m₁+m₂)]u₁ and v₂ = [2m₁/(m₁+m₂)]u₁. Special cases: if m₁ = m₂, v₁ = 0 and v₂ = u₁ (complete transfer of velocity, as in Newton's cradle); if m₂ ≫ m₁, v₁ ≈ −u₁ (bounces back) and v₂ ≈ 0.

For a perfectly inelastic collision, momentum conservation alone gives the common final velocity: v = m₁u₁/(m₁+m₂) when m₂ is initially at rest.