Question

Define the unit tangent vector, the principal unit normal vector, and the tangential and normal components of acceleration.

Short answer

Unit Tangent Vector - \( T(t) = \frac{r'(t)}{||r'(t)||} \), Principal Unit Normal Vector - \( N(t) = \frac{T'(t)}{||T'(t)||}\), Tangential Component of Acceleration - \( a_t = \frac{dv}{dt} \), Normal Component of Acceleration - \( a_n = r \frac{dv}{dt} \)

Step-by-step solution

Define Unit Tangent Vector

The unit tangent vector provides information about the direction of the path. For a given vector function \( r(t) \), the unit tangent vector \( T(t) \) at a particular point \( p \), is given by the derivative of \( r(t) \) at that point divided by its magnitude. Mathematically, \( T(t) = \frac{r'(t)}{||r'(t)||} \)

Define Principal Unit Normal Vector

The principal unit normal vector indicates the direction within the plane of the curve at point \( p \) and perpendicular to the tangent vector \( T \). It is obtained by differentiating the unit tangent vector \( T(t) \) and then dividing by its magnitude. It is denoted as \( N(t) \) and calculated as \( N(t) = \frac{T'(t)}{||T'(t)||}\) .

Define Tangential and Normal Components of Acceleration

The total acceleration vector of a particle is composed of the tangential acceleration and the normal (or radial) acceleration. The tangential component \( a_t \) determines the rate of change of speed along the path, and can be computed as \( a_t = \frac{dv}{dt} \) where \( v \) is speed. The normal component \( a_n \) signifies the rate of change of direction of the particle and can be computed as \( a_n = r \frac{dv}{dt} \), where \( r \) is the radius of curvature at that point.