Part 3.Determination of the Acceleration of a Particle when its Motion is described by the Natural Method

In the natural method of describing motion vector  is determined from its projections on a set of coordinate axes Mrnb whose origin is at M and who move together with the body. These axes, called the axes of the natural trihedron (or velocity axes), are directed as follows: axis Mr along the tangent to the path in the direction of the positive displacement S, axis Mn along the normal in the osculating plane towards the inside of the path, and axis Mb perpendicular to the former two to form a right-hand set. The normal Mn, which lies in the osculating plane (or in plane of the curve itself if the curve is two-dimensional), is called the principal normal, and the normal Mb perpendicular to it is called the binormal.

The acceleration of a particle lies in the osculating plane Mrn, hence its projection on the binormal is zero (ab=0).

Let the particle occupy a position M and have a velocity  at time t, and at time t1=t+ D t  let it occupy a position M 1 and have a velocity .

                           

From the theorem of the projection of a vector sum (or difference) on an axis we obtain:

                     

Noting that projections of vector on parallel axes are equal, draw through point M 1 axes Mr’ and Mn’ parallel to Mr and Mn, respectively, and denote the angle between the direction of vector and the tangent Mr (this angle is called the angle of contiguity).            

                                        

where k – the curvature of the curve at point M;

     R – the radius of curvature at point M.

We see that

Vr = V; Vn = 0;

                        V1 r = V1 cos D f; V1 n =V1 sin D f.

Hence

 

                             

 

When D t tends to zero, D f and D S tends to zero too and V 1 tends to V.

                                   

Then

                              

 

Multiplying the numerator and denominator of the fraction under the limit sign an by D f D S, we find

            ,

     since

               

     Finally we obtain

                                  

 

We have thus proved that the projection of the acceleration of a particle on the tangent to the path is equal to the first derivative of the numerical value of the velocity, or the second derivative of the displacement S, with respect to time; the projection of the acceleration on the principal normal is equal to the second power of the velocity divided by the radius of curvature of the path at the given point of the curve, the projection of the acceleration on the binormal is zero (ab=0).

The acceleration vector is the diagonal of a parallelogram constructed with the components and as its sides. As the components are mutually perpendicular, the magnitude of vector are given by the equation:

           .

The relations obtained express that the tangential component of the acceleration is equal to the rate of change of the speed of the particle, while the normal component is equal to the square of the speed divided by the radius of curvature of the path at point P.

Normal acceleration characterizes the change in direction of the velocity depending upon whether the speed of the particle increases or decreases, a_r positive or negative, and the vector component points in the direction of motion or against the direction of motion. The vector component , on the other hand, is always directed towards the center of curvature of the path.

The absolute or numerical value of the velocity is called the speed: speed is thus essentially a positive quantity.