Part 6/8:
\kappa = \left| \frac{d\mathbf{T}}{ds} \right|
]
Here, (s) denotes the arc length parameter. Re-expressing the derivative with respect to time involves the chain rule, leading to curvature's connection with velocity and normal components.
Geometric Visualization
The discussion emphasizes visualizing the particle's motion with a diagram showing:
The tangent vector pointing along the direction of motion.
The normal vector pointing toward the center of curvature.
The acceleration vector's components aligned with these vectors.
This geometric perspective clarifies how acceleration decomposes based on the particle's path shape and speed.