Work

Work: Definition and Mathematical Representation

Introduction

In physics, work refers to the energy transferred to or from an object via the application of force along a displacement. Work is a scalar quantity and depends on both the magnitude of the force and the displacement, as well as the angle between them.

Work links force and energy, making it one of the foundational concepts in classical mechanics.

Definition

The mathematical definition of work is:

$W=\overrightarrow{F}\cdot \overrightarrow{d}=|\overrightarrow{F}||\overrightarrow{d}|\cos \theta$

Where:

• $W$ is the work done,
• $\overrightarrow{F}$ is the applied force vector,
• $\overrightarrow{d}$ is the displacement vector,
• $\theta$ is the angle between the force and displacement vectors.

SI Unit

The SI unit of work is the joule (J):

$1\mathrm{J}=1\mathrm{N}\mathrm{\cdot }\mathrm{m}=1\mathrm{k}\mathrm{g}\mathrm{\cdot }{\mathrm{m}}^{\mathrm{2}}/{\mathrm{s}}^{\mathrm{2}}$

This means that one joule of work is done when a one-newton force moves an object one meter in the direction of the force.

Positive, Negative, and Zero Work

• Positive Work: Force is in the direction of displacement ($\theta <90^{\circ }$).
• Negative Work: Force is opposite to displacement ($\theta >90^{\circ }$).
• Zero Work: Force is perpendicular to displacement ($\theta =90^{\circ }$), or no displacement occurs.

Work Done by a Variable Force

If the force varies with position, work is computed using integration:

$W=\underset{x_1}{\overset{x_2}{\int }}F(x)dx$

This is common in systems with springs or non-uniform fields.

Work-Energy Theorem

The net work done on an object is equal to the change in its kinetic energy:

$W_{\text{net}}=\mathrm{\Delta }{E}_k=\frac{1}{2}m{v}^2-\frac{1}{2}m{u}^2$

Where:

• $m$ is mass,
• $v$ is final velocity,
• $u$ is initial velocity.

Applications

Work plays a vital role in:

• Mechanics (lifting, pulling, pushing)
• Engines and machines
• Thermodynamics (as energy transfer)
• Energy analysis and conservation

Examples

• Lifting a box: Positive work done by applied force.
• Friction during motion: Negative work done by frictional force.
• Carrying a bag horizontally: No work done if force is perpendicular to displacement.

See Also

• Energy
• Power
• Kinetic Energy
• Force
• Displacement
• Work-Energy Theorem