Work, Energy and Power

Work, Energy and Power

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Work is the scalar product of force and displacement: W = F d cosθ, where θ is the angle between the applied force and the displacement vector. It is measured in joules (J). A force perpendicular to motion does zero work; holding a dumbbell stationary produces no work because displacement is zero.

Energy is the capacity to do work. Two forms dominate MDCAT problems: kinetic energy K = ½ m v² (motion energy) and gravitational potential energy U = m g h with g = 9.8 m s⁻² near Earth’s surface. The work–energy theorem ties them together: net work done equals change in kinetic energy, W_net = ΔK.

Power is the time-rate of doing work, P = W / t, measured in watts (W); 1 W = 1 J s⁻¹. For a vehicle cruising at constant velocity, P = F v, so doubling speed doubles the power demand at the same tractive force.

High-yield pointers: (1) Always include g in P.E. (2) Use cosθ — a 60° push on a 10 N block over 5 m gives 25 J, not 50 J. (3) Power is scalar, not W (work).

---

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Defining Work Quantitatively

Work done by a constant force is W = F d cosθ. The displacement d must be in the same direction as the component of force considered. If θ = 0°, W = Fd (maximum); if θ = 90°, W = 0 (force perpendicular to motion — e.g., normal reaction on a sliding block does no work); if θ = 180°, W = −Fd (force opposing motion, such as kinetic friction). Work is path-independent only for conservative forces; for friction, W depends on the actual path length.

Kinetic Energy and the Work–Energy Theorem

A body of mass m moving at speed v possesses kinetic energy K = ½ m v². 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 = K_f − K_i = ½ m v_f² − ½ m v_i². This theorem converts force-based problems into energy bookkeeping, bypassing the need to compute accelerations explicitly.

Potential Energy

Gravitational P.E. near Earth’s surface: U_g = m g h, where h is height above a chosen reference. Elastic P.E. in a spring obeying Hooke’s law: U_s = ½ k x², where k is the spring constant (N m⁻¹) and x is extension/compression. Geometrically, this equals the triangular area under the straight-line F–x graph between 0 and x.

Conservation of Mechanical Energy

For a system with only conservative forces acting (gravity, ideal springs, electrostatic), the total mechanical energy E = K + U is constant. If friction, air drag, or any non-conservative force does work, mechanical energy decreases by exactly that amount — the lost energy becomes heat or sound.

Power

Average power P = W / t. Instantaneous power for a force acting on a moving body: P = F v cosθ. The SI unit watt equals 1 J s⁻¹; commercial electricity is billed in kilowatt-hours (1 kWh = 3.6 × 10⁶ J).

Efficiency

Real machines and engines always waste energy. Efficiency η = (useful output energy / input energy) × 100%, always less than 100%.

Common MDCAT Question Patterns

• Numerical: a 4 kg ball thrown upward at 20 m s⁻¹ — find max height using ½ m v² = m g h.
• Conceptual: comparing work done by gravity on two paths between the same heights.
• Graph-based: reading spring constant from a force–extension slope.
• Conceptual: which form of energy converts into which (chemical → kinetic → electrical).

---

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Conservative vs Non-Conservative Forces

A conservative force does work that depends only on initial and final positions, not on the path (gravity, ideal spring force, Coulomb). A non-conservative force (friction, air resistance, applied push) dissipates mechanical energy as heat; work done by it is path-dependent. The mechanical-energy equation with friction becomes: K_i + U_i = K_f + U_f + W_friction.

Variable Forces and the Area Interpretation

For a force that varies with position (such as a spring force F = −k x), work is the area under the F-versus-x curve. For a spring stretched from 0 to x, the graph is a triangle with base x and height kx, giving W = ½ k x² — identical to the stored elastic P.E. This integral definition, W = ∫ F · dx, extends the scalar-product idea to non-constant forces.

Power in Human Physiology and Machines

A 70 kg person climbing 3 m in 6 s does W = m g h = 2058 J, so P ≈ 343 W. Sustained human output is closer to 75–100 W; the rest is biological inefficiency, dissipated as heat. Car engines quote power in horsepower (1 hp ≈ 746 W). A vehicle requiring 800 N of tractive force at 30 m s⁻¹ needs P = 24 kW ≈ 32 hp to maintain that speed on a level road.

Worked Numeric Example

A 2 kg block slides down a frictionless 5 m incline from rest, then compresses a spring (k = 800 N m⁻¹) at the base. Find the maximum spring compression x. Energy conservation: m g h = ½ k x² (since incline height = 5 m, and final K = 0 at max compression). 2 × 9.8 × 5 = ½ × 800 × x² → 98 = 400 x² → x² = 0.245 → x ≈ 0.495 m.

Typical MDCAT Traps

1. Including or omitting g — P.E. without g is meaningless.
2. Confusing ½ m v² with ½ m v (linear momentum, not energy).
3. Assuming a constant force gives constant power — power scales with v.
4. Computing efficiency as a decimal instead of a percentage.
5. Believing mechanical energy is conserved whenever “energy is conserved” — global energy is conserved by the first law of thermodynamics; mechanical energy is only conserved when non-conservative work is zero.

Practice Prompts

1. A 1500 kg car travelling at 20 m s⁻¹ brakes uniformly to 10 m s⁻¹. Using the work–energy theorem, find the braking force if it stops in 25 m.
2. An electric motor rated 2 kW lifts a 100 kg load through 10 m. If the motor takes 7 s, calculate its efficiency.

---

Content adapted based on your selected roadmap duration. Switch tiers using the selector above.