For establishing the relationship between measurable and actual forces Merchant’s circle diagram will be used.

- Merchant circle diagram is used to analyze the forces acting in metal cutting.
- The analysis of three forces system, which balance each other for cutting to occur. Each system is a triangle of forces.

### Assumptions made in drawing Merchant’s circle:

- Shear surface is a plane extending upwards from the cutting edge.
- The tool is perfectly sharp and there is no contact along the clearance force.
- The cutting edge is a straight line extending perpendicular to the direction of motion and generates a plane surface as the work moves past it.
- The chip doesn’t flow to either side, that is chip width is constant.
- The depth of cut remains constant.
- Width of the too, is greater than that of the work.
- Work moves with uniform velocity relative tool tip.
- No built up edge is formed.

**The three triangles of forces in merchant’s circle diagram are**

- A triangle of forces for the cutting forces,
- A triangle of forces for the shear forces,
- A triangle of forces for the frictional forces.

Let F = Frictional force

N = Normal to frictional force

F

F

F

F

β = Friction angle

μ = Coefficient of friction = tanβ

F

N = Normal to frictional force

F

_{s}= Shear forceF

_{sn}= Normal to shear forceF

_{c}= Cutting force or tangential component of forceF

_{t}=Thrust force or feed forceβ = Friction angle

μ = Coefficient of friction = tanβ

F

_{c}and F_{t}are along and normal to the direction of velocity.
Let R = resultant force

Then resultant force is given by the formula

R = (F

_{c}^{2}+ F_{t}^{2})^{0.5 }
R = Diameter of merchant’s circle

F

_{t}, F_{c}forces are defined based on actual machining conditions- From the above merchant’s circle diagram it is found that there are three right angled triangles are present and all the three right angled triangle possessing common hypotenuse (largest side opposite to right angle in a right angled triangle).
- Merchant’s circle is used for establishing relationship between measurable and actual forces.