Single-point tool geometry: The “lip angle” (included cutting angle at the edge) is defined as which of the following?

Introduction / Context:
Understanding tool angles is central to predicting chip formation, tool strength, and surface finish. The lip angle (sometimes called the included cutting angle at the cutting edge) is a key geometric parameter of single-point tools that balances sharpness and robustness.

Given Data / Assumptions:

• Standard single-point turning tool with defined rake and relief faces.
• We consider the local geometry right at the cutting edge.
• Naming conventions: rake angle relates tool face to reference planes; relief angle separates flank from the work.

Concept / Approach:
The lip angle is formed by the intersection of the rake face (tool face) and the flank face at the cutting edge. It is therefore the angle between the tool face and the ground end surface of the flank. A larger lip angle strengthens the edge but increases cutting forces; a smaller lip angle sharpens the tool but risks chipping and rapid wear.

Step-by-Step Solution:

Verification / Alternative check:
Tool-grinding diagrams show the lip (included) angle as the complement of rake plus relief in a given reference, confirming it is the face–flank included angle at the edge, not a rake-to-plane relationship.

Why Other Options Are Wrong:

• (b) Describes a rake-angle definition relative to a base plane, not the included edge angle.
• (c) Refers to back rake relative to the machined surface tangent (a rake concept), not lip angle.
• (d) Incorrect since (a) is correct.
• (e) Not a standard definition for lip angle.

Common Pitfalls:
Confusing lip angle with rake or relief; overgrinding a small lip angle to “make it sharp,” which weakens the edge; ignoring material-specific recommendations for robust edge geometry.

Final Answer:
the angle between the tool face and the ground end surface of the flank (at the cutting edge)