A particular human hair has a Young\'s modulus of 4.0 109 N/m2 and a diameter of

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 180-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. .0508 Incorrect: Your answer is ____ cm

(c) If we think of the strand of hair as a spring, what is its spring constant? _____N m

(d) How does the hair's spring constant compare with that of ordinary springs in your physics laboratory? It is comparable. It is significantly larger. It is significantly smaller.

A beam is attached to a vertical wall with a hinge. The mass of the beam is 1000 kg and it is 5 m long. A steel support wire is tied from the end of the beam to the wall, making an angle of 30° with the beam (see the figure below). Recall the important values for steel: Young's modulus is
200 109 N/m2,
the shear modulus is
78 109 N/m2,
the bulk modulus is
140 109 N/m2,
the yield stress (elastic limit) is
290 106 N/m2,
and the ultimate breaking strength is
400 106 N/m2.

(e) By summing the torque about the axis passing through the hinge, calculate the tension in the support wire. Assume the beam is uniform so that the weight acts at its exact center.

______ N

(f) What is the minimum cross-sectional area of the steel wire so that it is not permanently stretched?

______cm2

Explanation / Answer

a) E=stress/strain
=>strain= L/L = stress/E

stress= F/A= ((180/1000)*9.81)/(3.14*150*150*10-12/4)
E=4x109 N/m2
=> L=23 * 0.02499363=0.57485 cm

b)Laluminium/Lhair = Ehair/Ealuminium
=>Laluminium= 0.57485*4/70=0.03284 cm

c) mg=kL
=> k= .180*9.81/0.57485 = 3.07175 N/cm = 307.175 N/m

d) It is larger.

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