Work, Energy, and Power

Introduction

We have explored how forces change an object’s velocity over a specific timeline. Now, we expand our diagnostic framework to observe what happens when a force acts on an object over a physical distance. Consider pulling a heavy luggage case across an airport terminal floor. You exert structural effort, your muscles burn chemical fuel, and the case speeds up.

In everyday conversation, we call this “hard work.” In physics, however, Work is a rigorous bookkeeping mechanism. It quantifies the precise mechanism by which energy is transferred from one distinct macroscopic store to another. Energy is not a physical object you can hold; it is a fundamental scalar property representing a system’s capacity to do work.

Key Concepts

1. Work Done by a Constant Force

Work is done only when a force causes a displacement along the same line of action as the force itself. If a force is applied at an angle to the direction of motion, only the parallel component of that vector performs mechanical work.

W = F * s * cos(θ)

Where: W = work done (Joules, J), F = magnitude of the force (N), s = displacement of the object (m), and θ = the angle between the force and displacement vectors.

• Positive Work (0° ≤ θ < 90°): The force has a component in the direction of motion, adding mechanical energy to the object (e.g., an engine pulling a train).
• Zero Work (θ = 90°): The force acts perpendicular to the displacement. It changes the object’s direction but does no work and transfers zero energy (e.g., the orbital centripetal gravitational pull on a satellite).
• Negative Work (90° < θ ≤ 180°): The force directly opposes motion, removing mechanical energy from the system and turning it into thermal energy (e.g., kinetic friction braking a vehicle).

2. Work Done by a Variable Force

If a force changes in magnitude as an object moves (such as compressing a spring), you cannot simply multiply the final force by the distance. On a Force–Displacement (F-s) graph, the total mechanical work done is represented by the total area enclosed underneath the curve.

3. Kinetic Energy and the Work-Energy Theorem

• Kinetic Energy (Ek): The energy a body possesses purely due to its state of motion. It is derived directly from the work required to accelerate a point mass from absolute rest to a target velocity v.

Ek = 0.5 * m * v²

• The Work-Energy Theorem: The net external mechanical work performed on a rigid body is completely identical to the net change in that body’s kinetic energy store.

Wnet = ΔEk = Ek_final - Ek_initial

4. Potential Energy Stores

Potential energy is energy stored within a system due to the spatial configuration or relative positions of its parts.

• Gravitational Potential Energy (Ep): Stored when a mass is lifted vertically against a uniform gravitational field.

  Ep = m * g * h

  Where: m = mass (kg), g = gravitational field strength (9.81 m s⁻²), and h = vertical change in height (m).

• Elastic Potential Energy (Eep): Stored when an elastic material is stretched or compressed. According to Hooke’s Law, the restoring force is proportional to the extension (F = k * x). The area under the F-x graph yields the stored energy:

  Eep = 0.5 * k * x²

  Where: k = spring constant (N m⁻¹) and x = extension/compression distance (m).

5. The Principle of Conservation of Energy

Energy can never be created out of nothing, nor can it ever be completely destroyed. It can only be transformed from one distinct store to another or transferred between parts of a system.

In a conservative system free from dissipative friction: Total Mechanical Energy = Ek + Ep = Constant Ek_initial + Ep_initial = Ek_final + Ep_final

In the presence of friction or fluid drag, mechanical energy is not destroyed, but rather dissipated (degraded) into a non-recoverable internal thermal energy store in the surroundings.

6. Power and Efficiency

• Power (P): The structural rate at which work is performed, or the rate at which energy is transferred over a time interval.

  P = W / t = ΔE / t (Units: Watts, W where 1 W = 1 J s⁻¹)

  When an object moves at a stable constant velocity v against a constant resisting force F, power simplifies to:

  P = F * v

• Efficiency (η): A dimensionless ratio tracking how effectively a device transfers useful energy without degrading it into useless ambient heat.

  Efficiency (η) = Useful Work Output / Total Work Input Efficiency (η) = Useful Power Output / Total Power Input

Common Mistakes

• Treating Work or Energy as a Vector: Because forces and displacements are vectors, students frequently try to assign spatial directions (like “left” or “down”) to energy values. Work and energy are pure scalar quantities. A negative sign on a work value does not mean “downwards”; it strictly indicates that energy is leaving the object’s kinetic store.
• Misinterpreting the Angle θ in the Work Equation: Students often automatically plug in whatever angle is written in the word problem without drawing a quick diagram. θ is strictly the angle wrapped between the force vector arrow and the displacement vector arrow. If a box is pulled horizontally along a floor by a rope angled at 30° above the horizontal, θ is 30°. But if it is lowered vertically down by a rope, look closely at the action paths.
• Using h as the Slanted Distance on Inclined Planes: When calculating gravitational potential energy changes (m * g * h) on a hill or ramp, students often use the length of the ramp. Remember, gravity only cares about vertical displacement. h must strictly be the vertical altitude change, calculated as: h = ramp_length * sin(θ).

Exam Tips

• The Core Conservation Blueprint: When analyzing tracks, rollercoasters, or pendulums, avoid kinematics if you can. Energy conservation is simpler. Set your lowest point as height h = 0. Then equate the total energy at position A to the total energy at position B.
• F-s Graph Analysis: If a Paper 2 question presents a graph showing Force changing over a Distance, do not try to find an average force to plug into a formula. Use a geometric calculation to find the area under the line. Count individual grid squares carefully if the line forms a complex curve.
• Convert Efficiency to Decimals Instantly: If an engine is described as “85% efficient”, immediately convert it to 0.85 in your scratchpad before setting up your output power equation. Keep track of whether a value is the absolute total input energy (large raw value from fuel) or the clean useful output energy (smaller payload value).

Practice Questions

Question 1 (Multiple Choice)

An electric motor raises a crate of mass m vertically upwards through a distance h at a constant, uniform speed v. The gravitational field strength is g. If the total electrical power input supplied to the motor is P, what is the overall mechanical efficiency of this system?

A. m * g * v / P B. m * g * h / P C. P / (m * g * v) D. P * v / (m * g * h)

Question 2 (Structured Paper 2 Style)

A spring-loaded toy launcher uses a compressed spring to project a small ball of mass 0.050 kg horizontally across a room. The spring has a characteristic spring constant k of 200 N m⁻¹ and is initially compressed by a distance of 0.080 metres from its equilibrium position. Assume all resistive forces and air drag within the barrel are entirely negligible.

(a) Calculate the total elastic potential energy stored in the compressed spring mechanism. [2 marks]

(b) Determine the maximum horizontal velocity of the ball at the exact instant it leaves the launcher muzzle and the spring returns to equilibrium. [2 marks]

(c) In a real-world trial, the ball leaves the muzzle with a velocity of 4.5 m s⁻¹. Calculate the efficiency of the launcher system during this transition. [2 marks]