Work, Energy, and Power

The concepts of work and energy are fundamental to all branches of physics and engineering. They provide a powerful alternative to Newton's laws for solving complex problems, especially those involving forces that change with position. While Newton's Laws are incredibly powerful, they can become mathematically cumbersome when forces are not constant or when the path of motion is complex. The Work-Energy theorem provides an elegant, alternative scalar approach. By tracking the transfer and transformation of energy, we can solve complex mechanics problems simply by comparing the initial and final states of a system, regardless of the path taken between them.

Work: The Transfer of Energy

Work: The Transfer of Energy Concepts

In physics, "work" has a very specific meaning. It is the transfer of energy to or from an object via the application of force along a displacement.

Work (WW)

The scalar product (dot product) of the force vector and the displacement vector. It represents the component of force acting in the direction of motion. The SI unit of work is the Joule (J), where 1 J=1 N⋅m1 \text{ J} = 1 \text{ N} \cdot \text{m}.

Work done by a Constant Force

Work done by a Constant Force Concepts

If a constant force F⃗\vec{F} is applied to an object that undergoes a straight-line displacement Δr⃗\Delta\vec{r}, the work done by that force is:

Constant Force Work

W=F⃗⋅Δr⃗=∣F⃗∣∣Δr⃗∣cos⁡(θ) W = \vec{F} \cdot \Delta\vec{r} = |\vec{F}| |\Delta\vec{r}| \cos(\theta)
Where:
  • F⃗\vec{F} is the constant force vector.
  • Δr⃗\Delta\vec{r} is the displacement vector.
  • θ\theta is the angle between the force and displacement vectors.

Important

Work can be positive, negative, or zero:
  • Positive Work (θ<90∘\theta < 90^\circ): The force is helping the motion, adding energy to the system.
  • Zero Work (θ=90∘\theta = 90^\circ): The force is perpendicular to the motion (e.g., normal force on a level surface). It transfers no energy.
  • Negative Work (90∘<θ≤180∘90^\circ < \theta \le 180^\circ): The force is opposing the motion (e.g., kinetic friction), removing energy from the system.

Work done by a Variable Force

Work done by a Variable Force Concepts

If the force changes magnitude or direction as the object moves, we must use calculus to find the work done. Work is the integral of the force along the path of motion.
W=∫xixfF(x) dx W = \int_{x_i}^{x_f} F(x) \, dx

Work done by a Variable Force Concepts

Graphically, this represents the area under the Force vs. Position curve.

Kinetic Energy and the Work-Energy Theorem

Kinetic Energy and the Work-Energy Theorem Concepts

Energy is the capacity to do work. The most basic form of mechanical energy is associated with motion.

Kinetic Energy (KK)

The energy an object possesses due to its motion. It is a scalar quantity (always positive or zero).
K=12mv2 K = \frac{1}{2}mv^2
The SI unit is the Joule (J).

Kinetic Energy and the Work-Energy Theorem Concepts

The connection between work and kinetic energy is codified in one of the most important theorems in mechanics.

The Work-Energy Theorem

The net work done by all forces acting on an object equals the change in its kinetic energy.
Wnet=ΔK=Kf−Ki=12mvf2−12mvi2 W_{net} = \Delta K = K_f - K_i = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2

Kinetic Energy and the Work-Energy Theorem Concepts

This theorem provides a powerful shortcut. If you know the initial and final speeds and the mass, you immediately know the total work done on the object, regardless of what forces were involved or the path taken.

Potential Energy and Conservative Forces

Potential Energy and Conservative Forces Concepts

Some forces, like gravity or an ideal spring, allow us to "store" work done as energy that can be recovered later. These are called conservative forces.

Conservative Force

A force is conservative if the work it does on an object moving between two points is independent of the path taken. Equivalently, the work done by a conservative force on an object moving around any closed path is zero. Examples: Gravity, Electrostatic force, Spring force.

Non-Conservative Force

A force that depends on the path taken. The work done by these forces cannot be recovered; it usually dissipates as heat or sound. Examples: Friction, Air resistance, Applied forces.

Potential Energy and Conservative Forces Concepts

For every conservative force, we can define a corresponding potential energy function (UU).

Potential Energy (UU)

Energy associated with the spatial configuration of a system of objects interacting via conservative forces. The change in potential energy is defined as the negative of the work done by the conservative force.
ΔU=−Wc=−∫F⃗c⋅dr⃗ \Delta U = -W_c = -\int \vec{F}_c \cdot d\vec{r}

Force from Potential Energy

Force from Potential Energy Concepts

Because the change in potential energy is the negative work done by a conservative force (ΔU=−Wc=−∫Fxdx\Delta U = -W_c = -\int F_x dx), we can reverse this relationship. If we know the potential energy function U(x)U(x), we can find the conservative force by taking the negative derivative:
Fx=−dUdx F_x = -\frac{dU}{dx}

Force from Potential Energy Concepts

In three dimensions, the conservative force vector is the negative gradient of the potential energy field: F⃗=−∇U\vec{F} = -\nabla U. This tells us that forces naturally push objects towards regions of lower potential energy.

Common Forms of Potential Energy

Common Forms of Potential Energy Concepts

  • Gravitational Potential Energy (UgU_g): Near the Earth's surface, Ug=mghU_g = mgh, where hh is the height above a chosen reference level (Ug=0U_g=0).
    • Elastic Potential Energy (UsU_s): Stored in a compressed or stretched spring following Hooke's Law (Fs=−kxF_s = -kx). Us=12kx2U_s = \frac{1}{2}kx^2, where xx is the displacement from equilibrium.

Conservation of Mechanical Energy

Conservation of Mechanical Energy Concepts

The total mechanical energy (EE) of a system is the sum of its kinetic and potential energies: E=K+UE = K + U.
If only conservative forces do work on a system, the total mechanical energy is conserved (remains constant).

Conservation of Mechanical Energy

Ei=Ef E_i = E_f Ki+Ugi+Usi=Kf+Ugf+Usf K_i + U_{gi} + U_{si} = K_f + U_{gf} + U_{sf}

Conservation of Mechanical Energy Concepts

If non-conservative forces (like friction) do work, the mechanical energy changes by an amount equal to that work:
Wnc=ΔE=Ef−Ei W_{nc} = \Delta E = E_f - E_i

Power

Power Concepts

While work tells us how much energy was transferred, power tells us how fast it was transferred. In engineering (e.g., designing motors or engines), power is often the critical limiting factor.

Power (PP)

The rate at which work is done or energy is transferred. The SI unit is the Watt (W), where 1 W=1 J/s1 \text{ W} = 1 \text{ J/s}.
Pavg=ΔWΔt=ΔEΔt P_{avg} = \frac{\Delta W}{\Delta t} = \frac{\Delta E}{\Delta t}
P(t)=dWdt P(t) = \frac{dW}{dt}

Power Concepts

For a constant force acting on an object moving with velocity v⃗\vec{v}, instantaneous power can be written as:
P=F⃗⋅v⃗ P = \vec{F} \cdot \vec{v}

Efficiency (η\eta)

Efficiency (η\eta) Concepts

No real machine transfers energy with 100% efficiency. Some energy is always lost to non-conservative forces like friction, typically converted into thermal energy. The mechanical efficiency is the ratio of useful power output to the total power input.
η=PoutPin=WoutWin \eta = \frac{P_{out}}{P_{in}} = \frac{W_{out}}{W_{in}}

Non-Conservative Forces and Dissipation

The Role of Friction

While conservative forces (like gravity and ideal springs) allow mechanical energy to be stored and perfectly recovered, non-conservative forces act differently. The most common non-conservative force in mechanical engineering is friction.
Friction always opposes motion. The work done by kinetic friction is inherently path-dependent and is always negative.

Work Done by Kinetic Friction

Calculates the mechanical energy dissipated as thermal energy due to friction.

$$ W_{f} = -f_k d = -(\mu_k N) d $$
Key Takeaways
  • Work (W=F⃗⋅d⃗W = \vec{F} \cdot \vec{d}) is energy transferred by a force acting over a distance. Positive work adds energy; negative work removes it.
  • The Work-Energy Theorem states that net work equals the change in kinetic energy (ΔK\Delta K).
  • Conservative forces (gravity, springs) allow the definition of Potential Energy (UU).
  • Mechanical Energy (E=K+UE = K + U) is conserved if only conservative forces do work. If friction is present, Wnc=ΔEW_{nc} = \Delta E.
  • Power is the rate of doing work or transferring energy (P=dWdtP = \frac{dW}{dt}).
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