As an example of our previous blogs, involving curvature, acceleration, and jerk, we now consider the example of the oscillation of a simple pendulum of length, l, which is subject only to gravitational acceleration.
The differential equation for the angle of deflection, phi, is
phi" - (g/l) sin phi 0,
where the primes denote differentiation with respect to time and g is the gravitational acceleration at the surface of the earth. Multiplying this equation by phi' and integrating once gives
phi'^2= (2g/l) cos phi,
where we have taken the constant of integration equal to the extreme angle of swing equal to 90 degrees, and, hence the cosine of this angle vanishes.
The velocity, acceleration and jerk are
v= (2gl)^1/2 cos^1/2 phi e_phi
a= -2g cos phi e_r - g sin phi e_phi
j= 3(2g^3/l)^1/2 {cos^1/2 phi sin phi e_r - cos^3/2 phi e_phi}
where e_r and e_phi are unit vectors in the radial and transverse directions. This has been reported by Shot in his article, "The time rate of change of acceleration".
Since r=l, a constant, the angular component of acceleration, l phi", does not vanish and hence the angular momentum per unit mass, h= l^2 phi', is not conserved. We will reach the same conclusion regarding the velocity space angular momentum, h = v^2 phi'.
Since the motion involves the jerk, the product of radii of curvature of the curve C and its polar reciprocal, C*, is
rho rho* = (a/v phi')^3.
Given that the radius of curvature of C is rho= a/phi', this expression reduces to the radius of curvature of the polar reciprocal, C*,
rho* = a^2/v^3 phi*.
The expression for the acceleration, a, can be obtained by squaring; we then obtain
a^2 = (v^2/l - g)^2 +(3/2) (v^2/l)g.
With the approximate equality of the centrifugal and gravitational accelerations, i. e., v^2/l ~ g, the acceleration will be their geometric mean,
a = {(v^2/l)g}^1/2= {v phi' g}^1/2,
omitting numerical coefficients, where v/l=phi', the angular frequency. So the radius of curvature of the polar reciprocal, C*, is related to the gravitational acceleration by
rho* = g'
The relation to the jerk has already been determined as
rho* = j v^2/h^2,
from the radial component of the jerk, assuming its normal component to vanish. Although this is not strictly true when h is not conserved, it should be a good approximation. Since, j= gv/l, And since v/l =phi'
rho* = j/phi' = g,
and the radius of curvature of the curve C, is
rho = a/phi'.
Gravity and jerk make themselves felt on the reciprocal polar curve C*, and are related through angular frequency, phi*
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