Mechanical Work

Core Definition

Mechanical Work ( $W$ ) is the scalar measure of energy transfer that occurs when an object is displaced by an external force:

W = \vec{F} \cdot \vec{d} = |\vec{F}| |\vec{d}| \cos\theta

where:

$\vec{F}$ is the constant force vector (in Newtons, $\text{N}$ ).
$\vec{d}$ is the displacement vector (in meters, $\text{m}$ ).
$\theta$ is the angle between the force vector and the direction of displacement.
$W$ is the resulting work scalar, measured in Joules ( $1\text{ J} = 1\text{ N}\cdot\text{m}$ ).

Concept Explanation

Because work is defined via the Dot Product, it is strictly a scalar quantity:

Positive Work ( $W > 0$ ): The force aids motion ( $\theta < 90^\circ$ ). Energy is added to the system.
Zero Work ( $W = 0$ ): The force acts perpendicular to motion ( $\theta = 90^\circ$ ). No mechanical work is performed (e.g., centripetal force in uniform circular motion, normal force on a level track).
Negative Work ( $W < 0$ ): The force opposes motion ( $\theta > 90^\circ$ ). Energy is removed from the system (e.g., kinetic friction, drag).

Straight-Line Displacement in 3D Space

If an object moves from initial point $P(x_1, y_1, z_1)$ to final point $Q(x_2, y_2, z_2)$ , the displacement vector is:

\vec{d} = \vec{PQ} = \langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \rangle

The work done by a constant 3D force

\vec{F} = \langle F_x, F_y, F_z \rangle

is computed as:

W = \vec{F} \cdot \vec{d} = F_x(x_2 - x_1) + F_y(y_2 - y_1) + F_z(z_2 - z_1)

Engineering Applications

Mechanical Conveyors & Actuators: Sizing motor output requirements to move industrial payloads across defined assembly tracks.
Wind Turbines & Aerodynamics: Calculating the power delivered by aerodynamic wind lift forces across rotating turbine blades.
Civil Earthmoving: Computing energy expended by bulldozers and cranes moving soil and building materials over elevations.

Example Problems

Example 1: 3D Coordinate Displacement Work

A force $\vec{F} = \langle 4, -4, 4 \rangle\text{ N}$ acts on a component as it moves from $P(-7, -2, 5)\text{ m}$ to $Q(8, -7, -5)\text{ m}$ . Compute the work done.

Solution

Compute Displacement Vector $\vec{d} = \vec{PQ}$ : $\vec{d} = \langle 8 - (-7), -7 - (-2), -5 - 5 \rangle = \langle 15, -5, -10 \rangle\text{ m}$
Compute Work via Dot Product: $W = \vec{F} \cdot \vec{d} = (4)(15) + (-4)(-5) + (4)(-10) = 60 + 20 - 40 = 40\text{ J}$

Connections & References

Parent Concepts: Dot Product | Dot and Cross Products
Related Notes: Vector Projections | Torque and Rotational Mechanics
Course Sessions: Calc-3 Session 03