Work Energy Power

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Work — Quantifying Force Through Distance:

Work is done when a force causes displacement in its own direction or any component thereof. W = F·s = Fs cosθ, where θ is the angle between the force vector and the displacement vector. SI unit: Joule (J). Work can be positive (force and displacement in the same general direction), zero (perpendicular), or negative (force opposes displacement). Examples: holding a heavy weight stationary does zero work on the weight (displacement = 0). Carrying a bucket horizontally around a room does zero work against gravity (mg is vertical, displacement is horizontal).

Kinetic Energy — The Energy of Motion:

KE = ½mv². This is always positive (or zero). Doubling the mass doubles KE; doubling the speed quadruples KE. This non-linear relationship explains why high-speed collisions are disproportionately destructive — a car at 100 km/h has four times the kinetic energy of the same car at 50 km/h.

Potential Energy — Stored Energy of Position:

Gravitational PE near Earth’s surface: U = mgh, where h is height above an arbitrary zero level. This is convenient for near-Earth problems but remember: h must be measured vertically, not along the inclined plane. Spring PE: U = ½kx², where x is displacement from the spring’s natural (unstretched) length.

Power and Efficiency:

Power = Work/time = Energy/time = P = W/t = J/s = Watt. Average power = F × v_average. Instantaneous power = F × v_instantaneous. 1 horsepower = 746 W. Efficiency = (useful power output / power input) × 100%. No machine has 100% efficiency due to dissipative forces (friction, air resistance).

⚡ CUET Tip: Work-Energy theorem W_net = ΔKE is your most versatile tool. It applies even when forces are non-conservative (like friction). Calculate the net work done by all forces, set it equal to the change in KE, and solve. Often easier than SUVAT when forces act over known distances.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Conservative Forces — Path Independence:

A force is conservative if the work done moving between two points is path-independent. Three equivalent conditions: (1) Work done in a closed loop = 0, (2) Work depends only on initial and final positions, (3) A potential energy function exists: F = -dU/dx. Gravity and spring forces are conservative. Friction is non-conservative — the work done by friction depends on the path length (longer path = more work dissipated as heat).

Mechanical Energy Conservation:

When only conservative forces do work (no friction, no air resistance), total mechanical energy E = KE + PE is constant. This allows powerful shortcuts: for a pendulum, the speed at any height depends only on that height, not the path taken. For a roller coaster, the maximum speed at the bottom of a dip depends only on the height difference, not the track shape.

⚡ CUET Tip: For a block sliding down a frictionless inclined plane from height h: v = √(2gh), regardless of the angle or length of the incline. This follows from energy conservation. Using kinematics with acceleration g sinθ would give the same result but requires more calculation.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Variable Force and Spring Systems:

For a force that varies with position, W = ∫F(x) dx from x₁ to x₂. The work done equals the area under the F-x graph. For a spring stretched from x = 0 to x = x₁: W = ½kx₁². For a spring compressed from x = 0 to x = x₁: W = -½kx₁² (negative because the force and displacement are in opposite directions — but magnitude of stored PE = ½kx₁²).

Elastic and Inelastic Collisions:

In an elastic collision, both momentum and kinetic energy are conserved. For a mass m₁ moving at u₁ colliding head-on with mass m₂ at rest:

• v₁ = ((m₁ - m₂)/(m₁ + m₂)) × u₁
• v₂ = (2m₁/(m₁ + m₂)) × u₁

In a perfectly inelastic collision (objects stick together), only momentum is conserved; kinetic energy is lost: KE_lost = ½μv_rel²(1 - e²) where μ = m₁m₂/(m₁ + m₂) is the reduced mass and e is the coefficient of restitution.

Power-Velocity Relationship:

For a vehicle of mass m climbing a slope with constant power P: the engine provides power P = F_engine × v. At steady speed, F_engine = mg sinθ + friction. For level ground (θ = 0): v_max = P/friction. When friction is proportional to v (air resistance ∝ v² at high speeds), the vehicle approaches terminal velocity where engine force equals drag.

⚡ CUET Pattern: CUET frequently tests mechanical energy conservation in roller coaster or pendulum-type problems. A classic CUET question: a pendulum bob of mass m is released from horizontal position (height = L, the string length). At the bottom, its speed v = √(2gL). The tension in the string at the bottom is T = mg + mv²/L = mg + 2mg = 3mg. The centripetal force at the bottom is mv²/L = 2mg upward, in addition to the weight mg, giving total tension 3mg. At the top of the swing (inverted position), T = mg - mv²/L (if it has enough speed to complete the circle), and minimum speed at the top = √(gL), giving minimum tension = 0. If the bob doesn’t have enough energy to complete the circle, it oscillates.