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)

**Calculating the magnitude of force when its application is shown using the figure below:**

U =

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

= _{t} dS

= F_{t} – S

Where,

F_{t} – S = Area under the curve

**Calculate the magnitude of force:**

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})

**Links of Previous Main Topic:-**

- Introduction to statics
- Introduction to vector algebra
- Two dimensional force systems
- Introduction concept of equilibrium of rigid body
- Friction introduction
- Introduction about distributed forces
- Area moments of inertia in rectangular and polar coordinates
- Mass moment of inertia introduction
- Work done by force
- Kinematics of particles
- Position vector velocity and acceleration
- Plane kinematics of rigid bodies introduction
- Combined motion of translation and rotation
- Rectilinear motion in kinetics of particles

**Links of Next Mechanical Engineering Topics:-**