Kinetic and Potential Energy

🟢 Lite — Quick Review (1h–1d)

Work is energy transferred when a force causes displacement in its direction: W = F·d·cos θ (F in newtons, d in metres, θ = angle between F and d). Kinetic energy: Eₖ = ½mv². Gravitational potential energy: Eₚ = mgh (h measured from chosen reference point). Spring potential energy: Eₚ = ½kx². Power: P = W/t = F·v (watts). Efficiency: η = (W_out/W_in) × 100%. High-yield SNBT pointers: (1) When θ = 90° → W = 0 (force perpendicular to motion does no work). (2) cos θ determines sign: positive if θ < 90°, negative if θ > 90°. (3) Mechanical energy Eₘ = Eₖ + Eₚ is conserved only when non-conservative forces (friction, drag) are absent.

🟡 Standard — Regular Study (2d–2mo)

Definition of Work

Work done by a constant force is defined as the scalar product W = F · d = Fd cos θ. A force does positive work when its component acts in the direction of displacement (θ < 90°), negative work when it opposes motion (θ > 90°), and zero work when perpendicular (θ = 90°). This scalar nature means direction matters only through the angle — unlike force, work has no component perpendicular to displacement.

Kinetic and Potential Energy

Kinetic energy Eₖ = ½mv² depends on mass and the square of velocity — doubling speed quadruples kinetic energy. Gravitational potential energy Eₚ = mgh depends on height h relative to a chosen zero level; changing the reference point changes the numerical value but not physical results in a consistent system. Elastic potential energy stored in a compressed or stretched spring follows Eₚ = ½kx², where k is the spring constant.

Conservation and the Work-Energy Theorem

The Work-Energy Theorem states: net work done on an object equals its change in kinetic energy: W_net = ΔEₖ = Eₖ(final) − Eₖ(initial). In a closed system with only conservative forces (gravity, spring force), total mechanical energy Eₘ = Eₖ + Eₚ remains constant: Eₘ₁ = Eₘ₂. Friction converts mechanical energy into thermal energy, breaking conservation of Eₘ.

Power

Power P = W/t measures how fast work is done or energy is transferred. The alternate form P = Fv applies when force and velocity are parallel — useful for engine or motor ratings. Efficiency η compares useful output work to input work.

Common Exam Patterns

SNBT Saintek combines work-energy analysis with projectile motion and inclined planes. Expect questions asking for speed at a certain height, minimum work to reach a point, or power output given force and velocity data. Always verify whether non-conservative forces are present before applying energy conservation.

🔴 Extended — Deep Study (3mo+)

Reference Point Dependence in Potential Energy

Gravitational potential energy Eₚ = mgh is reference-dependent — setting h = 0 at ground versus at the launch point yields different numerical values. However, differences in potential energy ΔEₚ = mgΔh remain reference-independent, making physically meaningful results independent of your choice. Spring potential energy ½kx² similarly depends on the equilibrium position as its natural zero.

Variable Forces and Non-Conservative Systems

When force varies with position, work equals the area under the F-vs-displacement graph. For non-conservative forces (friction, air resistance), use W_net = ΔEₖ directly, or include the work done by friction: W_total = ΔEₖ + W_friction. Energy is still conserved in the broader sense (friction energy becomes heat), but mechanical energy is not conserved.

Worked Micro-Example

A 2 kg block slides down a frictionless 30° incline from height h = 5 m. Initial Eₘ = Eₚ = mgh = 2 × 10 × 5 = 100 J at top. At bottom, Eₚ = 0, so Eₖ = 100 J. Solving ½mv² = 100 gives v = √(100) = 10 m/s. If kinetic friction coefficient μₖ = 0.2 acts over the 10 m incline length: W_friction = −μₖmg cos 30° × 10 ≈ −34.6 J. Then Eₖ(bottom) = 100 − 34.6 = 65.4 J, giving v = √(65.4) ≈ 8.1 m/s.

Common Mistakes to Avoid

Using the angle between force and the horizontal instead of between force and displacement vector.
Assuming energy conservation when friction or applied external forces are present.
Forgetting that Eₖ depends on v² — a common trap in multi-object problems.

Practice Prompts

A 3 kg object is thrown upward at 20 m/s. Using energy conservation (no air resistance), find the maximum height reached. Answer: h = (20²)/(2×10) = 20 m.
A motor lifts a 500 kg load at constant speed 2 m/s. If efficiency is 80%, what input power is required? Hint: Output P = mgv = 500 × 10 × 2 = 10,000 W; input = 10,000/0.80 = 12,500 W.

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Kinetic and Potential Energy