The "lead" in pencils is a graphite composition with a Young's modulus of about 1 ✕ 10⁹ N/m². Calculate the change in length (in mm) of the lead in an automatic pencil if you tap it straight into the pencil with a force of 4.9 N. The lead is 0.70 mm in diameter and 55 mm long.

0.701 mm.

Explanation:

From Hook's law,

Young modulus = Stress/Strain.

Stress = Force/area = F/A

Strain = ΔL/L.

Therefore,

γ = (F/A)/(ΔL/L)

γ = FL/ΔLA Equation 1

Where γ = Young's  modulus, F = Force, L = Length of the pencil, ΔL = change in length of the pencil, A = cross sectional area.

Make ΔL the subject of the equation,

ΔL = FL/γA Equation 2

But,

A = πd²/4 Equation 3

Where d = diameter

Substitute equation 3 into equation 2

ΔL = 4FL/γπd² Equation 4

Given: F = 4.9 N, L = 55 mm = 0.055 m, d = 0.7 mm = 0.0007 m, γ = 1×10⁹ N/m², π = 3.14.

Substitute into equation 4

ΔL = (4×4.9×0.055)/(1×10⁹×3.14×0.0007²)

ΔL = 1.078/1538.6

ΔL = 7.01×10⁻⁴ m.

ΔL  = 0.701 mm.

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