A particular human hair has a Young\'s modulus of 4.0 ? 10 9 N/m 2 and a diamete

Question

A particular human hair has a Young's modulus of  4.0 ? 109 N/m2  and a diameter of 150 ?m.

(a) If a 250-g object is suspended by the single strand of hair that is originally 23 cm long, by how much will the hair stretch? _________cm

(b) If the same object were hung by an aluminum wire of the same dimensions as the hair, by how much would the aluminum stretch? (Try to do this part without repeating the previous calculation, but use proportional reasoning instead. Also, see the table.) The Young's modulus of aluminum is

A particular human hair has a Young's modulus of 4.0 ? 109 N/m2 and a diameter of 150 ?m. (a) If a 250-g object is suspended by the single strand of hair that is originally 23 cm long, by how much will the hair stretch? _________cm (b) If the same object were hung by an aluminum wire of the same dimensions as the hair, by how much would the aluminum stretch? (Try to do this part without repeating the previous calculation, but use proportional reasoning instead. Also, see the table.) The Young's modulus of aluminum is 70 ? 109 N/m2. _______________cm (c) If we think of the strand of hair as a spring, what is its spring constant?____________N/M

Explanation / Answer

a. Youngs modulus Y = Fx/ADx

where F = force = mg, x= original length, A= area = pi r^2, Dx= change of length, Y = youngs modulus

Dx = Fx/AY

Dx =0.25*9.8* 0.23/(3.15*75*75*10^-12*4*10^9)

Dx=7.97 mm

b. if aluminium is used,

Dx = 0.250*9.8*0.23/3.14* 75*75*10^-12*70*10^9)

Dx=0.455 mm

spring constant and elongation are related by F= kx

k = 0.25*9.8*/0.455 mm

k =5384.61 N/m

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