The Shape of a Ponytail

1. The Shape of a Ponytail Shazeaa Ishmael, Department of Physics Redefining the equation of state: Figure 2 b shows the predicted shape of a ponytail at different lengths with two different the equation of states; , which was the empirical form found simply by studying a ponytail of length 25cm. However, it is obvious there are some discrepancies between the shape the theory predicts and the measured ponytails. To create a better fit to the experimental data, better definitions of the currently constant values of π0 and R* are required. This was achieved by analysis of the data on the mean squared radial excursion, r02(s), and curvature, k02(s), as functions of arc length for individual hairs, displayed in Figure 3. Redefining the equation of state: Background Information: The close examination of the physical structure of a ponytail was prompted by the rapidly growing hair care industry. The familiar shape of the ponytail arose from properties of the fibres themselves and how they interact with each other. An equation was proposed for the shape of an asymmetric bundle of hairs, uniformly distributed across the cross section and all of the same length. Fig 1: Image of a switch of N~9500 fibres, approximately 25 cm long. Denotes the coordinate system for the envelope shape R(s) in terms of the arc length s(z).[1] The Current Equation : Equation 1 is the potential energy contained within an asymmetric ponytail, where N is the number of fibres, A is a bending modulus of the hair, ≈ 8x10-2Nm2 and λ is the linear mass density, ≈65μg/cm, The first term is the curvature field, or the energy contained in the initial splaying of the ponytail, the second term is gravitational potential energy, even though it looks complex is it simply the weight of the fibres below a certain point, the final term is confinement energy. When considering the hairs in a ponytail, it is best to think of them as a fibre confined within a cylinder, representative of the surrounding hairs in the ponytail. Minimising equation 1 using the Euler-Lagrange gives equation 2. When l= (A/λg) 1/3also termed the equation of state. By solving this forth order differential equation the envelope of the ponytail R(s) can be found. Fig2b is the shape that equation 2 predicts. What follows summarises the research I carried out to improve the equation of state and hence the current model. Fig 3:Mean squared radial excursion and curvature as functions of arc length, from processing stereoscopic image pairs. The reconstructed arc lengths cluster tightly around 24.50 ±0.05 cm (inset histogram), demonstrating the accuracy of the image processing and analysis methodology. Error bars are standard errors from ensemble averaging (N = 115 fibres in total). [1] Using Mathematica the data was plotted and a functional form found but trying different fits. The most accurate forms found were r0(s) =r1(s/L*)1/2 and k02=k1(s/L) , where L*=25cm, r1 = and k1=, .These are can now be used to find more correct forms of R*and π0which are related as follows; R*=R1 r0(s) and π0 =π1 k02(s)/R*. These functional forms were then substituted in the original equation of state and using Mathematica, the equation was solved , the program created an image of the envelope of the ponytail at different overall lengths. Fig 4 below contains this image. Fig 4: Predicted profile of a ponytail at 5cm, 10cm, 15cm, 20cm and 25 cm.. Fig 2: Ponytails trimmed to different lengths, a) experimentally measured samples cut from 25cm at intervals of 5cm b) the predicted profiles from equation 2[1] The theory and the experimental data are certainly in agreement with each other at length greater than 10cm. More research can be carried out to improve the fit, especially at shorter lenghts. Areas of particular interest are the effect of hair ends in the ponytail, the effect of the hairband and the shape is the hair is actually attached to the scalp References:[1]R. Goldstein, P. Warren, R. Ball, Physical Review Letters 108, 078101 (2012)