Question

A particular human hair has a Young's modulus of 3.77×109 N/m2and a diameter of 146 μm.

If a 256 g object is suspended by the single strand of hair that is originally 15.5 cmlong, by how much ΔL hair

will the hair stretch?

If the same object were hung from an aluminum wire of the same dimensions as the hair, by how much ΔLAl

would the aluminum stretch?

If the strand of hair is modeled as a spring, what is its spring constant khair?

Answer 1

Given,

Young's Modulus, Y = 3.77 * 10⁹ N/m²

The diameter, d = 146 m = 146 * 10^-6 m

So, radius, r = 146 * 10^-6 / 2

= 73 * 10^-6 m

The mass, m = 256 g = 0.256 Kg

The length, L = 15.5 cm = 0.155 m

a)

The length, L = FL / AY

L = mgL / r² Y

= 0.256 * 9.8 * 0.155 /   * (73 * 10^-6)² * 3.77 * 10⁹

= 6.16 * 10^-3 m

b)

Young's Modulus of Aluminum, Y_Al = 69 * 10⁹ N/m²

L = mgL / r² Y_Al

= 0.256 * 9.8 * 0.155 /   * (73 * 10^-6)² * 69 * 10⁹

= 3.39 * 10^-4 m

c)

The spring constant, k = YA / L

k = Yr² / L

k = 3.77 * 10⁹ * * (73 * 10^-6)² / 0.155

= 407.12 N/m