**Subject :**Elements of Mechanical Engineering

## Kinematics of a particle moving on a curve

**Kinematics of a particle moving on a curve:**** **Suppose a particle is moving on a curve. Then its velocity is given by

where represents the speed along the path and is the unit vector tangent to the path. The acceleration becomes

Now

.............................(1)

**Some concepts from differential geometry: Osculating Plane:**

Let us consider a curve. The tangents at P and Q are shown. . The plane containing both the tangents, when is called osculating plane.

Osculate is a fancy word for kiss. The osculating plane just kisses the path. The path may break out of this plane later, but for now, the path is turning within the osculating plane.

In the above figure, the plane containing the unit tangents at P and Q will become an osculating plane as . Osculating plane may change from point to point or may remain constant.