Work and conservation of energy

The component along the displacement

For a constant force on a particle, W = Fd cos θ. F is the force magnitude in N; d is the displacement magnitude in m; θ is the angle between force and displacement. The product F cos θ is the component parallel to the displacement. A perpendicular component does not contribute to this work, even if it changes the direction of velocity.

Work is measured in joules: 1 J = 1 N·m. It is positive when the force component is along the displacement and negative when it opposes it. The sign expresses the direction of transfer in the energy model, not the presence of human effort. A force can exist without doing work over a particular section.

If the force changes along a straight path, divide the path into small sections, estimate parallel F times displacement in each one and add them. On a graph of parallel F against x, this sum corresponds to the signed area. For an ideal spring, whose force grows linearly with deformation, the area is triangular rather than rectangular.

Why does mv²/2 appear?

Consider a constant net force along the motion of a mass m. By the second law, net F = ma. Constant-acceleration kinematics gives v² − v₀² = 2aΔx. Multiplying by m/2 gives maΔx = mv²/2 − mv₀²/2. The left-hand side is the net work.

We define translational kinetic energy as K = mv²/2. Thus, net W = ΔK. Although the short derivation uses a simple situation, the theorem applies more generally to a particle within Newtonian mechanics. For extended bodies with rotation and deformation, the corresponding forms of energy must be included.

Adding the work contributions is essential. If the pulling force does +80 J and friction −80 J, ΔK = 0, even though real transfers occur. The theorem does not say that the work of any individual force is ΔK. Nor does it directly give the direction of velocity: K depends on speed squared and is a scalar.

Potential energy and conservative forces

A force is conservative when its work between two points depends only on the endpoints. A potential energy U can then be defined such that conservative W = −ΔU. Gravity and the force of an ideal spring are examples. In contrast, frictional work generally depends on the path and cannot be represented by a single potential energy of position.

Near Earth’s surface, with g almost constant, gravitational U = mgh. In this lesson use g = 9,8 m/s². The reference h = 0 is chosen for convenience; changing that reference adds the same constant to the values and does not change ΔU. Potential energy belongs to the configuration of the body–Earth system, even when we shorten the wording to “the body’s energy”.

For an ideal spring obeying F = −kx, x is the deformation relative to its equilibrium length and k is the spring constant in N/m. The work needed to deform it quasi-statically is the triangular area on an applied-force–deformation graph: elastic U = kx²/2. This relationship requires deformations within the linear elastic regime.

Building a conservation equation

We separate net W = conservative W + other W. Since conservative W = −ΔU, we obtain ΔK + ΔU = other W. If the work of the other forces is zero, K + U stays constant. This is conservation of mechanical energy within the chosen model.

A frictionless ramp can have a large normal force and still conserve mechanical energy, because the normal is perpendicular to the displacement at every point and does no work on the particle. The shape of the ramp can change the forces and descent time without changing the final speed for the same height difference and initial energy.

The equality initial K + initial U = final K + final U must be accompanied by its assumptions. Negligible air resistance, a fixed support and no other transfers are examples. Do not mistake “frictionless” for an automatic guarantee: an external motor may still supply energy to the system.

When mechanical energy decreases

Friction can transform mechanical energy into internal energy of the bodies in contact. For a constant kinetic friction force opposing sliding through a distance d, the work is −fd. The total energy of a sufficiently broad system remains conserved, but K + U may decrease.

Write an explicit account: initial mechanical energy + energy supplied = final mechanical energy + increase in internal energy + other outputs. Not all the internal energy generated stays in one body; its distribution depends on the materials and conditions. Saying only “there was a loss” should mean a loss of useful mechanical energy in the model, not destruction of energy.

Do not count the same transfer twice. You can treat friction as external work on the body, or include the surfaces in the system and track internal energy. The choice of boundary determines the terms. A physically clear equation is more important than always following a formula with the same names.

Power, efficiency and checks

Average power is average P = transferred E/Δt, in watts. When a force and the velocity of its point of application are known, instantaneous power is P = Fv cos θ. The second relationship shows that the same force can transfer energy at different rates depending on velocity.

Efficiency η is useful energy/energy received, or useful power/power received when compared in the same operating regime. It has no unit and is usually expressed as a percentage. An efficiency of 75% means that, for every 100 J received, 75 J correspond to the output defined as useful; the rest has not disappeared.

Check units, sign and scale. Translational energy cannot be negative; potential energy can, depending on the reference. A negative square-root argument usually indicates an impossible assumption or an accounting error. If height is missing, look for height differences; if time is missing, energy may be the shortest route; if the question asks for duration, you may still need dynamics.

Worked examples, without skipping steps

Follow the reasoning first. Then cover the solution and try to reconstruct it in your own words.

Now try it yourself

There is no timer. Write what you know, choose a relationship and justify each step. Hints and solutions are available whenever you need them.

Mistakes worth avoiding

A large force always does a lot of work.

Displacement and the parallel component must be considered.

Potential energy must have its zero at ground level.

Zero is a chosen reference; differences are independent of it.

With friction, energy is no longer conserved.

Mechanical energy can decrease while internal energy increases.

Power and energy have the same unit.

Energy is J; power is J/s or W.