Editing 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:

<math>
W = \vec{F} \cdot \vec{d} = |\vec{F}| |\vec{d}| \cos\theta
</math>

Where:
* <math>W</math> is the work done,
* <math>\vec{F}</math> is the applied force vector,
* <math>\vec{d}</math> is the displacement vector,
* <math>\theta</math> is the angle between the force and displacement vectors.

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

<math>
1\, \mathrm{J} = 1\, \mathrm{N \cdot m} = 1\, \mathrm{kg \cdot m^2 / s^2}
</math>

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 (<math>\theta < 90^\circ</math>).
* '''Negative Work''': Force is opposite to displacement (<math>\theta > 90^\circ</math>).
* '''Zero Work''': Force is perpendicular to displacement (<math>\theta = 90^\circ</math>), or no displacement occurs.

== Work Done by a Variable Force ==
If the force varies with position, work is computed using integration:

<math>
W = \int_{x_1}^{x_2} F(x) \, dx
</math>

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:

<math>
W_{\text{net}} = \Delta E_k = \frac{1}{2}mv^2 - \frac{1}{2}mu^2
</math>

Where:
* <math>m</math> is mass,
* <math>v</math> is final velocity,
* <math>u</math> 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]]