Position vector, vector representation of plane curves, unit tangent, unit normal, curvature, curvilinear motion, velocity, acceleration, tangential and normal components of acceleration

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POSITION VECTOR, VECTOR REPRESENTATION OF PLANE CURVES, UNIT TANGENT, PRINCIPAL NORMAL, CURVATURE, CURVILINEAR MOTION, VELOCITY, ACCELERATION, TANGENTIAL AND NORMAL COMPONENTS OF ACCELERATION

Def. Position vector (or radius vector). The vector

running from the origin to a point P(x, y) in the plane is called the position vector or radius vector of P. The vectors and are unit vectors along the positive the x and y axes respectively.

Plane curves. We can think of a curve in the plane as a path of a moving point. The definition of a plane curve is essentially an analytical implementation of this view. A plane curve is defined parametrically as the graph of parametric equations

1) x = x(t)

y = y(t)

where the functions x(t) and y(t) are continuous and the range of the parameter t is some interval (finite or infinite) of the real axis. In vector language a plane curve is defined as the graph of points traced out by a position vector

where the functions f and g are continuous and the range of the parameter t is some interval (finite or infinite) of the real axis. The positive direction of a space curve is the direction of increasing t. i.e Δs/Δt > 0.

If we interpret t as time, 1) can be regarded as defining the path of a moving point. The point may pass through the same point in the plane several times which means that the curve may intersect itself. This definition that we have given gives a curve that is very general and one that may not be very smooth. For example, it could include things like the track of a tiny particle in Brownian movement over a long period of time (a very random path going first in one direction and then abruptly changing to another direction — randomly and suddenly changing from one direction to another).

Derivative of a vector. The derivative of a vector

with respect to t is given by

The derivative is a vector tangent to the curve at the point in question. See Fig. 2. If the variable t represents time, then represents the velocity with which the terminal point of the radius vector describes the curve. Similarly, dv/dt represents its acceleration a along the curve.

Unit tangent vector. Let a curve C be given by

R(s) = x(s) i + y(s) j

where the parameter s represents the arc length measured from some fixed point on the curve. Then the unit tangent vector to the curve at a particular point P is given by

T = dR/ds

That this is so can be seen from Fig. 3 which shows R and R + ΔR at points P and P'. The quotient ΔR/Δs is a vector along the line of the chord PP'. Since the length of ΔR is the length of the chord PP', we see that when P' approaches P the limit of the length of ΔR/Δs is unity. Furthermore, the limiting direction of PP' is that of the tangent at P. Therefore

Differentiation of R with respect to t gives

Note that if t represents time then ds/dt is speed and dR/dt is velocity. From 2) we get

From T = dR/ds we obtain

Principal normal. The principal normal, denoted by N , at a point P on a curve C, is a unit vector in the direction of dT/ds (providing dT/ds is not zero, in which case the principal normal is not defined). The principal normal is necessarily perpendicular to the unit tangent vector T. It points toward the concave side of the curve. See Fig. 4. From 4) above we obtain

Curvature of a curve at point P. The curvature κ of a curve at a point P is given by the magnitude of dT/ds:

Thus

7) dT/ds = κN

Radius of curvature. The radius of curvature ρ is the reciprocal of the curvature:

8) ρ = 1/κ

Center of curvature. The tip of the vector ρN, drawn from point P as initial point, is called the center of curvature of the curve at point P.

Velocity and acceleration in curvilinear motion. Consider a point P(x, y) moving along a curve defined by the parametric equations

x = x(t)

y = y(t)

where t is time. The position vector of the moving point is given by

The velocity is given by

The magnitude of is given by

where v_x and v_y are the x and y components of the velocity i.e.

v_x = dx/dt

v_y = dy/dt .

The direction of at P is along the tangent to the curve at P.

The acceleration is given by

The magnitude of is given by

where a_x and a_y are the x and y components of the acceleration i.e.

a_x = d²x/dt²

a_y = d²y/dt² .

Tangential and normal components of acceleration. First note that

Thus

which gives the resolution of the acceleration vector at P in the directions of the unit tangent T and principal normal N. Denoting these components in the directions of T and N by and we have

where is the radius of curvature of the curve at P.

Since

we have

as a second way of determining .