Work Formula: W = F·d (Calculus & Physics)

Work

The physics term for the amount of energy required to move an object over a given path subject to a given force.

Work $\displaystyle = \int_a^b f(x)\,dx\text{,}$ where $f(x)$ = the force acting in the direction of or opposite to movement.
or $\displaystyle \int_a^b \mathbf{f}(\mathbf{x})\cdot\mathbf{dx}\text{,}$ where $\mathbf{f}(\mathbf{x})$ = the vector function for force acting at the vector-valued point $\mathbf{x}\text{.}$

Key Formula

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

Where:

$W$ = Work done, measured in joules (J) or foot-pounds
$F$ = Magnitude of the applied force
$d$ = Distance the object moves
$\theta$ = Angle between the force vector and the direction of motion

Worked Example

Problem: A person pushes a crate with a constant force of 50 newtons across a floor for 8 meters. The force is applied in the same direction the crate moves. How much work is done?

Step 1: Identify the known values: force is 50 N, distance is 8 m, and the angle between the force and the direction of motion is 0°.

F = 50 \text{ N}, \quad d = 8 \text{ m}, \quad \theta = 0°

Step 2: Apply the work formula. Since the force is in the same direction as the motion, cos 0° = 1.

W = 50 \cdot 8 \cdot \cos 0° = 50 \cdot 8 \cdot 1

Step 3: Multiply to find the work done.

W = 400 \text{ J}

Answer: The work done on the crate is 400 joules.

Why It Matters

Work connects force and energy, making it central to problems in physics and applied mathematics. In calculus, work is computed using definite integrals when force varies along a path, such as stretching a spring or pumping water out of a tank. Understanding work also lays the groundwork for concepts like kinetic energy and potential energy.

Common Mistakes

Mistake: Forgetting the angle and assuming all applied force contributes to work.

Correction: Only the component of force in the direction of motion does work. If you push at an angle θ to the direction of movement, you must include the cos θ factor. Pushing perpendicular to the motion (θ = 90°) does zero work.

Related Terms

Integral — Used to compute work when force varies
Cosine — Projects force onto direction of motion