Work, Energy, and Power

Work, Energy, and Power

$Learning Objectives$

Define work, kinetic energy, potential energy, and power.
Calculate the work done by constant and variable forces.
Apply the Work-Energy Theorem to relate net work to changes in kinetic energy.
Distinguish between conservative and non-conservative forces.
Use the principle of Conservation of Mechanical Energy to solve mechanics problems.
Calculate power output and mechanical efficiency for engineering systems.

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.

$W$

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 $1 \text{ J} = 1 \text{ N} \cdot \text{m}$ .

Work done by a Constant Force

$Work done by a Constant Force Concepts$

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

Work done by a Constant Force

Calculates the work done by a constant force acting over a straight-line displacement.

W = \vec{F} \cdot \Delta\vec{r} = |\vec{F}| |\Delta\vec{r}| \cos(\theta)

Variables

Symbol	Description	Unit
$W$	Work done	J
$\vec{F}$	Constant force vector	N
$\Delta\vec{r}$	Displacement vector	m
$\theta$	Angle between force and displacement vectors	$^\circ$

$Sign of Work Done$

Work can be positive, negative, or zero:

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

$Work and force-angle visual$

Only the component of force parallel to displacement contributes to translational work.

Block displacement vector with an angled applied force and its parallel projection. — Work as a force-displacement dot product
The projection F cos theta is the part of the force that transfers energy through the displacement.

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.

Work done by a Variable Force

Calculates the work done by a variable force along a 1D path.

W = \int_{x_i}^{x_f} F(x) \, dx

Variables

Symbol	Description	Unit
$W$	Work done	J
$F(x)$	Variable force as a function of position	N
$x_i$	Initial position	m
$x_f$	Final position	m

$Work done by a Variable Force Concepts$

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

$Variable-force work visual$

For a force that varies with position, work is the signed area under the force-displacement curve.

Force versus displacement graph with the area under a varying force curve shaded. — Work from an F-x graph
The integral of force with respect to displacement gives work; regions below the axis contribute negative work.

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.

$K$

The energy an object possesses due to its motion. It is a scalar quantity (always positive or zero). The SI unit is the Joule (J).

Kinetic Energy

Calculates the translational kinetic energy of an object.

K = \frac{1}{2}mv^2

Variables

Symbol	Description	Unit
$K$	Kinetic energy	J
$m$	Mass of the object	kg
$v$	Speed of the object	$\text{m/s}$

$Kinetic Energy and the Work-Energy Theorem Concepts$

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.

The Work-Energy Theorem

Relates the net work done on an object to its change in kinetic energy.

W_{net} = \Delta K = K_f - K_i = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2

Variables

Symbol	Description	Unit
$W_{net}$	Net work done on the object	J
$\Delta K$	Change in kinetic energy	J
$K_f$	Final kinetic energy	J
$K_i$	Initial kinetic energy	J
$m$	Mass of the object	kg
$v_f$	Final speed	$\text{m/s}$
$v_i$	Initial speed	$\text{m/s}$

$Kinetic Energy and the Work-Energy Theorem Concepts$

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 for which the work between two states cannot be represented by a single position-only potential-energy function. Such interactions can transfer mechanical energy into internal, thermal, acoustic, or other forms. Examples: kinetic friction and aerodynamic drag.

$Potential Energy and Conservative Forces Concepts$

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

$U$

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.

Change in Potential Energy

Defines the change in potential energy as the negative work done by a conservative force.

\Delta U = -W_c = -\int \vec{F}_c \cdot d\vec{r}

Variables

Symbol	Description	Unit
$\Delta U$	Change in potential energy	J
$W_c$	Work done by conservative force	J
$\vec{F}_c$	Conservative force vector	N
$d\vec{r}$	Differential displacement vector	m

Force from Potential Energy

$Force from Potential Energy Concepts$

Because the change in potential energy is the negative work done by a conservative force ( $\Delta U = -W_c = -\int F_x dx$ ), we can reverse this relationship. If we know the potential energy function $U(x)$ , we can find the conservative force by taking the negative derivative:

Force from Potential Energy

Calculates the 1D conservative force as the negative derivative of the potential energy function.

F_x = -\frac{dU}{dx}

Variables

Symbol	Description	Unit
$F_x$	Conservative force in x-direction	N
$U$	Potential energy function	J
$x$	Position	m

$Force from Potential Energy Concepts$

In three dimensions, the conservative force vector is the negative gradient of the potential energy field: $\vec{F} = -\nabla U$ . 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 ( $U_g$ ): Near the Earth's surface, $U_g = mgh$ $U_{g} = m g h$ , where $h$ $h$ is the height above a chosen reference level ( $U_g=0$ $U_{g} = 0$ ).
- Elastic Potential Energy ( $U_s$ ): Stored in a compressed or stretched spring following Hooke's Law ( $F_s = -kx$ ). $U_s = \frac{1}{2}kx^2$ , where $x$ is the displacement from equilibrium.

Conservation of Mechanical Energy

$Conservation of Mechanical Energy Concepts$

The total mechanical energy ( $E$ ) of a system is the sum of its kinetic and potential energies: $E = K + U$ .

If only conservative forces do work on a system, the total mechanical energy is conserved (remains constant).

$Conservation of Mechanical Energy$

If only conservative forces do work on a system, the total mechanical energy is conserved (remains constant).

Conservation of Mechanical Energy

States that total mechanical energy remains constant when only conservative forces act.

E_i = E_f

K_i + U_{gi} + U_{si} = K_f + U_{gf} + U_{sf}

Variables

Symbol	Description	Unit
$E$	Total mechanical energy	J
$K$	Kinetic energy	J
$U_g$	Gravitational potential energy	J
$U_s$	Elastic potential energy	J
$i$	Initial state	-
$f$	Final state	-

$Interactive Simulation: Conservation of Energy$

Use this energy model to watch kinetic and potential energy trade places while total mechanical energy stays tracked.

2 kg

10 m

Energy Diagnostic Chart

Kinetic Energy (K)0 J

Potential Energy (U)196 J

Total Mechanical Energy (E)196 J

Governing Formulas

Conservation of Energy

E = K + U = \text{constant}

Kinetic & Potential Energy

K = \frac{1}{2}mv^2, \quad U = mgh

Time0.0 s

Height (h)10.0 m

Speed (v)0.0 m/s

Position (x)0.0 m

$Conservation of Mechanical Energy Concepts$

If non-conservative forces (like friction) do work, the mechanical energy changes by an amount equal to that work:

Work Done by Non-Conservative Forces

Calculates the change in mechanical energy caused by non-conservative forces.

W_{nc} = \Delta E = E_f - E_i

Variables

Symbol	Description	Unit
$W_{nc}$	Work done by non-conservative forces	J
$\Delta E$	Change in total mechanical energy	J
$E_f$	Final total mechanical energy	J
$E_i$	Initial total mechanical energy	J

$Energy-transfer visual$

Track energy as a conserved accounting system: mechanical energy can shift between kinetic and potential forms, while non-conservative processes transfer part of it into thermal or other internal energy.

Three stacked energy states showing gravitational potential, kinetic, and thermal energy shares. — Energy transfer and dissipation
A stacked energy view makes conversion and dissipation explicit without implying that thermal energy disappears.

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.

$P$

The rate at which work is done or energy is transferred. The SI unit is the Watt (W), where $1 \text{ W} = 1 \text{ J/s}$ .

Average and Instantaneous Power

Calculates the rate of energy transfer or work done.

P_{avg} = \frac{\Delta W}{\Delta t} = \frac{\Delta E}{\Delta t}

P(t) = \frac{dW}{dt}

Variables

Symbol	Description	Unit
$P_{avg}$	Average power	W
$P(t)$	Instantaneous power	W
$\Delta W$	Work done	J
$\Delta E$	Energy transferred	J
$\Delta t$	Time interval	s

$Power Concepts$

For a constant force acting on an object moving with velocity $\vec{v}$ , instantaneous power can be written as:

Instantaneous Power from Force and Velocity

Calculates instantaneous power when a constant force is applied to an object moving at a certain velocity.

P = \vec{F} \cdot \vec{v}

Variables

Symbol	Description	Unit
$P$	Instantaneous power	W
$\vec{F}$	Force vector	N
$\vec{v}$	Velocity vector	$\text{m/s}$

$Interactive Simulation: Power and Work Rate$

Adjust force, displacement, time, and force direction to connect work transfer with power demand.

θ = 30°

150 N

8.0 m

12 s

30 deg

0.45

Governing Equations

Work Equation

W = F \cdot d \cdot \cos(\theta)

Average Power

P = \frac{W}{t}

Total Work Transferred (W)

1,039.23 J

Avg Power Rate (P)

86.60 W

Parallel force component (F·cosθ)

129.90 N

Perpendicular force (F·sinθ)

75.00 N

Efficiency ( $\eta$ )

$\eta$

No real machine converts all input energy into the intended useful output. Part of the input is transferred into non-useful forms such as thermal energy, vibration, or sound. Mechanical efficiency is the ratio of useful power output to total power input.

Mechanical Efficiency

Calculates the ratio of useful power or work output to the total input.

\eta = \frac{P_{out}}{P_{in}} = \frac{W_{out}}{W_{in}}

Variables

Symbol	Description	Unit
$\eta$	Mechanical efficiency	-
$P_{out}$	Useful power output	W
$P_{in}$	Total power input	W
$W_{out}$	Useful work output	J
$W_{in}$	Total work input	J

$Efficiency visual$

Efficiency is an energy or power accounting ratio, not a new source of energy: useful output plus losses must trace back to the input.

Sankey-style input energy flow splitting into useful output and losses. — Efficiency as an energy-flow ratio
The useful branch divided by the total input gives efficiency, while the loss branch accounts for non-useful transfers.

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.

Kinetic friction opposes relative sliding at the contact. Its work on a selected body is negative when the friction force is opposite that body's displacement, as in the sliding-block cases used here; in other systems, such as a moving belt accelerating an object, friction can do positive work on that object.

Work Done by Kinetic Friction

Calculates friction work for the common case where kinetic friction is opposite the displacement.

W_{f} = -f_k d = -(\mu_k N) d

Variables

Symbol	Description	Unit
$W_{f}$	Work done by friction	J
$f_k$	Kinetic friction force	N
$d$	Distance traveled	m
$\mu_k$	Coefficient of kinetic friction	-
$N$	Normal force	N

Key Takeaways

Work ( $W = \vec{F} \cdot \vec{d}$ ) 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 ( $\Delta K$ ).
Conservative forces (gravity, springs) allow the definition of Potential Energy ( $U$ ).
Mechanical Energy ( $E = K + U$ ) is conserved if only conservative forces do work. If friction is present, $W_{nc} = \Delta E$ .
Power is the rate of doing work or transferring energy ( $P = \frac{dW}{dt}$ ).

$References$

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Learning Objectives

Work: The Transfer of Energy

Work: The Transfer of Energy Concepts

Work (WWW)

Work done by a Constant Force

Work done by a Constant Force Concepts

Work done by a Constant Force

Sign of Work Done

Work and force-angle visual

Work done by a Variable Force

Work done by a Variable Force Concepts

Work done by a Variable Force

Work done by a Variable Force Concepts

Variable-force work visual

Kinetic Energy and the Work-Energy Theorem

Kinetic Energy and the Work-Energy Theorem Concepts

Kinetic Energy (KKK)

Kinetic Energy

Kinetic Energy and the Work-Energy Theorem Concepts

The Work-Energy Theorem

The Work-Energy Theorem

Kinetic Energy and the Work-Energy Theorem Concepts

Potential Energy and Conservative Forces

Potential Energy and Conservative Forces Concepts

Conservative Force

Non-Conservative Force

Potential Energy and Conservative Forces Concepts

Potential Energy (UUU)

Change in Potential Energy

Force from Potential Energy

Force from Potential Energy Concepts

Force from Potential Energy

Force from Potential Energy Concepts

Common Forms of Potential Energy

Common Forms of Potential Energy Concepts

Conservation of Mechanical Energy

Conservation of Mechanical Energy Concepts

Conservation of Mechanical Energy

Conservation of Mechanical Energy

Interactive Simulation: Conservation of Energy

Energy Diagnostic Chart

Conservation of Mechanical Energy Concepts

Work Done by Non-Conservative Forces

Energy-transfer visual

Power

Power Concepts

Power (PPP)

Average and Instantaneous Power

Power Concepts

Instantaneous Power from Force and Velocity

Interactive Simulation: Power and Work Rate

Efficiency (η\etaη)

Efficiency (η\etaη) Concepts

Mechanical Efficiency

Efficiency visual

Non-Conservative Forces and Dissipation

The Role of Friction

Work Done by Kinetic Friction

References

$Learning Objectives$

$Work: The Transfer of Energy Concepts$

$W$

$Work done by a Constant Force Concepts$

$Sign of Work Done$

$Work and force-angle visual$

$Work done by a Variable Force Concepts$

$Work done by a Variable Force Concepts$

$Variable-force work visual$

$Kinetic Energy and the Work-Energy Theorem Concepts$

$K$

$Kinetic Energy and the Work-Energy Theorem Concepts$

$The Work-Energy Theorem$

$Kinetic Energy and the Work-Energy Theorem Concepts$

$Potential Energy and Conservative Forces Concepts$

$Conservative Force$

$Non-Conservative Force$

$Potential Energy and Conservative Forces Concepts$

$U$

$Force from Potential Energy Concepts$

$Force from Potential Energy Concepts$

$Common Forms of Potential Energy Concepts$

$Conservation of Mechanical Energy Concepts$

$Conservation of Mechanical Energy$

$Interactive Simulation: Conservation of Energy$

$Conservation of Mechanical Energy Concepts$

$Energy-transfer visual$

$Power Concepts$

$P$

$Power Concepts$

$Interactive Simulation: Power and Work Rate$

Efficiency ( $\eta$ )

$\eta$

$Efficiency visual$

$The Role of Friction$

$References$