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 Newtonmeter (Nm). For the force and displacement calculation, we use the unit of work as Nm. (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
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