Work, Energy, and Power

14.1 Why Energy Matters in Physics

Energy is one of the most important concepts in all of science. It appears in every branch of physics — mechanics, thermodynamics, electromagnetism, quantum mechanics, and relativity. It is conserved: the total energy of an isolated system never changes. It takes many forms — kinetic, potential, thermal, chemical, nuclear, electromagnetic — and converts among them. These two features — conservation and multiple forms — make energy an extraordinarily powerful analytical tool.

In mechanics, the energy perspective provides a completely different way to analyze motion from the force perspective. Instead of asking “what forces act on this object and how do they produce acceleration?” Newton’s approach, the energy approach asks “how much work was done on this object and how did that change its kinetic energy?” Both approaches give the same answer, but each is more convenient in different situations.

The energy approach is especially powerful when forces are complicated, variable, or unknown in detail — as in collisions, spring systems, and problems where objects travel along curves. The kinematic and force methods require knowing force at every instant; energy methods require only knowing the total work done, regardless of the path taken.

Albert Einstein, paraphrase of the first law of thermodynamics

14.2 What Physicists Mean by Work

In everyday language, “work” means any effort or activity. In physics, work has a precise definition that does not always match everyday usage. Physical work is done on an object when a force acts on it and the object moves in the direction of the force, or has a component of displacement in the direction of the force.

Work is a scalar quantity — it has magnitude and sign but no direction. It is measured in joules (J), where 1 J = 1 N·m = 1 kg·m²/s². Work transfers energy from one object or system to another.

The physical work done equals the force applied times the displacement in the direction of the force. If the force and displacement are parallel, W = Fd. If they are at an angle, only the component of force along the displacement does work.

14.3 Conditions Required for Work

Three conditions must all be met for a force to do work on an object:

• A force must act on the object.
• The object must undergo a displacement; it must move.
• There must be a component of the force along the direction of displacement; the force cannot be purely perpendicular to the motion.

If any of these conditions is not satisfied, the force does zero work on the object. This leads to some counterintuitive results: gravity does no work on an object moving horizontally, the normal force does no work on an object sliding on a horizontal surface, and a centripetal force does no work because it is always perpendicular to velocity.

14.4 Positive, Negative, and Zero Work

Work can be positive, negative, or zero, depending on the angle between the force and the displacement.

• Positive work (θ < 90°): the force has a component in the direction of displacement. The force transfers energy to the object, increasing its kinetic energy. Example: an engine pushing a car forward.
• Negative work (θ > 90°): the force has a component opposite to the displacement. The force removes energy from the object. Example: friction opposing a sliding box; gravity acting on a rising object.
• Zero work (θ = 90°): the force is perpendicular to the displacement. No energy is transferred. Example: normal force on a horizontally moving box; centripetal force on an orbiting satellite.

The sign of work tells you the direction of energy transfer: positive work adds energy to the object; negative work removes it.

14.5 The Work Equation

When a constant force F acts on an object that undergoes displacement d, with angle θ between the force vector and displacement vector, the work done is:

W = Fd cosθ

The cosine factor extracts the component of force along the displacement. When θ = 0°, force and displacement are parallel and W = Fd. When θ = 90°, the force is perpendicular and W = 0. When θ = 180°, the force is antiparallel and W = -Fd.

14.6 Energy as the Capacity to Do Work

Energy is the capacity of a system to do work. A moving car has energy and can exert forces over a distance, such as crumpling metal in a crash. A raised weight has energy and can do work as it falls. A compressed spring has energy and can push objects when released.

Energy is conserved: it cannot be created or destroyed, only transferred between systems or converted between forms. This law — the law of conservation of energy — is one of the most fundamental laws in all of physics. Work is the mechanism by which energy is transferred: when a force does work on an object, it transfers energy to or from that object.

The SI unit of energy is the joule (J = N·m = kg·m²/s²). This is also the unit of work, because work measures the amount of energy transferred.

14.7 Kinetic Energy

Kinetic energy (KE) is the energy an object possesses due to its motion. It depends on both mass and velocity:

KE = ½mv²

Kinetic energy is a scalar — it has no direction. It depends on speed, the magnitude of velocity, not velocity itself. Two objects moving in opposite directions at the same speed have the same kinetic energy.

The velocity dependence is quadratic: doubling the speed quadruples the kinetic energy. This is why collision safety is so much more important at high speeds: a car at 100 km/h has four times the kinetic energy of the same car at 50 km/h, meaning it needs four times the stopping distance for the same braking force and absorbs four times the energy in a crash.

14.8 The Work-Energy Theorem

The work-energy theorem is one of the most powerful results in mechanics. It states: the net work done on an object equals the change in the object’s kinetic energy.

W_net = ΔKE = KE_f - KE_i = ½mv_f² - ½mv_i²

This theorem follows directly from Newton’s second law combined with kinematics: applying F = ma and the kinematic equation v_f² = v_i² + 2aΔx yields W_net = FΔx = ½mv_f² - ½mv_i². The work-energy theorem is not a new law; it is a consequence of Newton’s second law expressed in energy language.

The theorem is useful because it bypasses the need to know the acceleration or the time — only the net work and the velocities matter. It is especially powerful for variable forces, curved paths, and problems where the force changes in magnitude or direction along the path.

14.9 Power and Rate of Doing Work

Power is the rate at which work is done, or equivalently, the rate at which energy is transferred. Two machines that do the same total work differ in power if one does it faster than the other.

P = W / t

An equivalent expression in terms of force and velocity: when a constant force F acts on an object moving at velocity v in the direction of the force:

P = Fv

14.10 Mechanical Power in Real Systems

In real machines, not all input power appears as useful output power. Friction, air resistance, and other dissipative forces convert some mechanical energy to thermal energy. The efficiency of a machine quantifies this:

efficiency = W_out / W_in = P_out / P_in

No real machine is 100% efficient. Gasoline engines are typically 20-35% efficient; most energy becomes heat. Electric motors can be 85-95% efficient. The human body is about 25% efficient at converting food energy to mechanical work.

The P = Fv form is particularly useful for vehicles. The maximum speed of a car occurs when the engine’s power equals the power dissipated by drag and friction: P_engine = F_drag × v_max.

14.11 Energy Transfer and Transformation

Work is one mechanism for transferring energy between objects. When a force does positive work on an object, energy is transferred from the agent exerting the force to the object. When a force does negative work, energy flows the other direction.

Energy also transforms between forms without being transferred between objects. A falling ball converts gravitational potential energy to kinetic energy. A braking car converts kinetic energy to thermal energy. A battery converts chemical energy to electrical energy. In every transformation, the total energy is conserved; only its form changes.

The work-energy theorem is the mechanics version of energy conservation: all energy transferred into an object as net work appears as kinetic energy. Chapter 15 extends this to include potential energy stored in gravitational and spring systems, leading to the full conservation of mechanical energy principle.

14.12 Common Mistakes in Work and Energy Problems

Chapter Summary

Work is the transfer of energy by a force; the work-energy theorem links the net work on an object to its change in kinetic energy. Power is the rate of work done.

• Work: W = Fd cosθ; units = joules (J = N·m). Positive when force has component along displacement; negative when opposing; zero when perpendicular.
• Three conditions for work: a force must act, the object must move, and the force must have a component along the displacement.
• Kinetic energy: KE = ½mv²; scalar; always ≥ 0; quadrupled when speed doubles.
• Work-energy theorem: W_net = ΔKE = ½mv_f² - ½mv_i².
• Power: P = W/t; units = watts (W = J/s). Also P = Fv for a constant force along velocity.
• Efficiency = P_out/P_in; always ≤ 1; no real machine is 100% efficient.
• Energy is conserved; work is the mechanism of transfer; kinetic energy is one of its forms.

Key Terms

work

The energy transferred to or from an object by a force acting over a displacement: W = Fd cosθ; SI unit: joule (J).

joule (J)

The SI unit of work and energy: 1 J = 1 N·m = 1 kg·m²/s².

kinetic energy (KE)

Energy possessed by an object due to its motion: KE = ½mv²; scalar; always ≥ 0.

work-energy theorem

The net work done on an object equals its change in kinetic energy: W_net = ΔKE.

power

The rate at which work is done or energy is transferred: P = W/t; SI unit: watt (W = J/s).

watt (W)

The SI unit of power: 1 W = 1 J/s = 1 kg·m²/s³.

efficiency

The ratio of useful output energy or power to total input energy or power; dimensionless; always ≤ 1.

energy

The capacity of a system to do work; conserved in all physical processes; SI unit: joule (J).

positive work

Work done when the force has a component in the direction of displacement, 0° ≤ θ < 90°; increases the object’s kinetic energy.

negative work

Work done when the force has a component opposing displacement, 90° < θ ≤ 180°; decreases the object’s kinetic energy.

Concept Review Questions

1. Define work in the physics sense. Give two examples from everyday life where effort is expended but no physical work is done.
2. What three conditions must be met for a force to do work on an object?
3. A centripetal force acts on a satellite in circular orbit. Does it do work? Explain.
4. In the equation W = Fd cosθ, what does the angle θ represent? What happens to work as θ increases from 0° to 180°?
5. Explain in your own words what the work-energy theorem says. How does it differ from Newton’s second law?
6. A car moving at 100 km/h has how many times the kinetic energy of the same car at 50 km/h? Explain why.
7. What is the difference between work and power? Give an example where one person does more work but another has greater power.
8. What does it mean for a machine to be 80% efficient? Where does the other 20% of energy go?
9. Friction does negative work on a sliding box. What happens to that energy — is it destroyed?
10. Can an object have kinetic energy and zero velocity simultaneously? Can it have zero kinetic energy and non-zero velocity?

Practice Problems

Chapter 14 Checkpoint