Work, Energy and Power

What this chapter is about

This chapter introduces three of the most fundamental quantities in physics: work, energy and power. You have already studied motion and forces in earlier chapters. Now you learn how to connect force and displacement through a single number called work, and how that work changes the energy of a body. Energy is a quantity that can be stored, transferred and transformed, but the total energy of an isolated system stays constant. This principle, called the conservation of energy, runs through every branch of physics.

You will meet kinetic energy (energy due to motion) and potential energy (energy due to position or configuration). The chapter also shows how quickly energy is transferred, which is measured by power. These ideas let you solve problems that would be very hard using Newton's laws alone — for example, finding the speed of a roller-coaster car at the bottom of a slope without tracking every instant of the motion.

By the end you should be able to calculate work done by constant and variable forces, apply the work-energy theorem, use conservation of mechanical energy, and distinguish between conservative and non-conservative forces. You will also see how collisions are analysed using momentum and energy together.

Key ideas

• Work done by a constant force F on a body that moves through displacement s is W = F s cos θ, where θ is the angle between F and s. Work is a scalar; its SI unit is the joule (J), equal to 1 N m.
• Work can be positive (force aids motion), negative (force opposes motion) or zero (force is perpendicular to motion, as in uniform circular motion where the centripetal force does no work).
• Kinetic energy of a body of mass m moving with speed v is K = (1/2) m v². It is always non-negative.
• The work-energy theorem states that the net work done on a body equals the change in its kinetic energy: W_net = K_f − K_i.
• Potential energy is energy stored due to position or configuration. Gravitational potential energy near Earth's surface is U = mgh (taking the reference level where U = 0). Elastic potential energy of a spring stretched or compressed by x from its natural length is U = (1/2) k x², where k is the spring constant.
• A conservative force is one for which the work done depends only on initial and final positions, not on the path. Gravity and the spring force are conservative; friction is non-conservative because it dissipates energy as heat.
• Mechanical energy E = K + U is conserved when only conservative forces act. When non-conservative forces (like friction) act, mechanical energy decreases.
• Power is the rate of doing work: P = W/t for constant power, or P = dW/dt in general. Also P = F v cos θ. The SI unit is the watt (W), equal to 1 J/s.

Formulas and facts to remember

• Work by constant force: W = F s cos θ (F is magnitude, s is displacement magnitude, θ is angle between them).
• Work by variable force along a straight line: W = ∫ F dx from x_i to x_f.
• Kinetic energy: K = (1/2) m v².
• Gravitational potential energy (near Earth): U = mgh (h above chosen reference).
• Elastic potential energy: U = (1/2) k x² (x is extension or compression).
• Work-energy theorem: W_net = ΔK = (1/2) m v_f² − (1/2) m v_i².
• Conservation of mechanical energy (no non-conservative forces): K_i + U_i = K_f + U_f.
• Power: P = W/t (average), P = dW/dt (instantaneous), P = F v (when F is along v).
• 1 horsepower ≈ 746 W.

Worked examples

### Example 1 — Work done by gravity and a normal force

A 5 kg block slides 4 m down a smooth inclined plane that makes 30° with the horizontal. Find the work done by (a) gravity, (b) the normal force.

Solution

Take g = 10 m/s².

(a) Gravity acts vertically downward. The vertical drop is h = 4 sin 30° = 4 × 0.5 = 2 m. Work by gravity W_g = mgh = 5 × 10 × 2 = 100 J. (Alternatively, W = F s cos θ with F = 50 N, s = 4 m, θ = 60° gives the same result: 50 × 4 × 0.5 = 100 J.)

(b) The normal force is perpendicular to the displacement along the plane, so cos 90° = 0. Work by normal force = 0 J.

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### Example 2 — Using the work-energy theorem

A car of mass 1200 kg moving at 20 m/s brakes uniformly and stops after travelling 50 m on a level road. Find the average braking force.

Solution

Initial kinetic energy K_i = (1/2) × 1200 × 20² = 240 000 J. Final kinetic energy K_f = 0 (car stops). Work done by braking force W = K_f − K_i = 0 − 240 000 = −240 000 J.

Let the braking force be F (opposing motion, so work is negative). W = −F × 50 m −240 000 = −F × 50 F = 4800 N.

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### Example 3 — Conservation of energy with a spring

A 0.2 kg ball is placed against a horizontal spring (k = 500 N/m) compressed by 0.1 m and then released. The surface is frictionless. Find the speed of the ball when it leaves the spring.

Solution

Initially the ball is at rest, so K_i = 0. Elastic potential energy stored U_i = (1/2) k x² = (1/2) × 500 × (0.1)² = 2.5 J. When the spring reaches its natural length, U_f = 0 and K_f = (1/2) m v².

By conservation of mechanical energy: K_i + U_i = K_f + U_f 0 + 2.5 = (1/2) × 0.2 × v² + 0 v² = 2.5 / 0.1 = 25 v = 5 m/s.

Common mistakes

• Forgetting the cos θ factor and always writing W = F s → Remember that only the component of force along the displacement does work.
• Using mgh even when the reference level is not stated → Choose a clear reference level and state it; potential energy is defined relative to that level.
• Believing kinetic energy can be negative → Kinetic energy is (1/2) m v²; since m > 0 and v² ≥ 0, kinetic energy is never negative.
• Applying conservation of mechanical energy when friction acts → Friction is non-conservative; mechanical energy is not conserved. Use work-energy theorem and include the work done by friction.
• Confusing energy (joules) with power (watts) → Energy is the total work done; power is how fast it is done.

Quick revision

• Work = force component along displacement × displacement; unit is joule.
• Work-energy theorem: net work = change in kinetic energy.
• Mechanical energy (K + U) is conserved when only conservative forces act.
• Potential energy depends on the choice of reference level.
• Power = energy transferred per unit time; unit is watt (1 W = 1 J/s).
• In collisions, momentum is always conserved; kinetic energy is conserved only in elastic collisions.