Work can be defined as the application of force to produce a displacement in the direction of force. It is a scalar product.

Mathematically,

dU = .

or,

|| = dS

From Fig. 6.1, the above equation can be written as,

dU = F dS cos

= (F cos ) dS

= F_{t} dS (|_{t}|= F cos )

**Note:**

From the above equation, work can also be defined as product of force in the direction of its application and displacement in the same direction. In other words, work is the product of component of force and its displacement.

See Fig. 6.1, dS and F is at right angle (90^{0}). Thus, force F_{n} will be zero as it does no work. Consider the component of force F_{t}, the value of work done will be positive if F_{t} will act on the same direction of displacement.

Similarly, the value of work done will be negative if F_{t} will act on the opposite direction of displacement.

**Unit of Work:**

S.I. unit of work is Joule (J) or Newton-meter (N-m). For the force and displacement calculation, we use the unit of work as N-m. (Joule will be used in terms of energy)

U =

= _{x} dx + F_{y}dy + F_{z} dz)

= _{t} dS

= F_{t} – S

Where,

F_{t} – S = Area under the curve

To calculate the force, it can be written as,

U_{1 – 2} =

=

= ½ K (x_{2}^{2} – x_{1}^{2})

** Note:**

Since the work is done on the body, its sign will be negative.

Again,

**Calculate the magnitude of force as shown in figure below**

U_{1 – 2} =

=

=

=

=

Hence,

Work done will be,

U_{1 – 2} = ½ m (v_{2}^{2} – v_{1}^{2})