When you stretch a wire, rope, or rubber band, it gets thinner as well as longer

Question

When you stretch a wire, rope, or rubber band, it gets thinner as well as longer. When Hooke's law holds, the fractional decrease in width is proportional to the tensile strain. If w0 is the original width and w is the change in width, then w/w0=l/l0, where the minus sign reminds us that width decreases when length increases. The dimensionless constant, , different for different materials, is called Poisson's ratio.

Part A: A steel rod 2.0 m long has a cross-sectional area of 0.30cm2. The rod is now hung by one end from a support structure and a 550-kg milling machine is hung from the rod's lower end. If the steel rod has a circular cross section and a Poisson's ratio of 0.23, what is its change in diameter when the milling machine is hung from it?

Part B: A cylinder made of nickel (Poisson's ratio = 0.42) has radius 2.0 cm. What tensile force F must be applied perpendicular to each end of the cylinder to cause its radius to decrease by 0.10 mm? Assume that the breaking stress and proportional limit for the metal are extremely large and are not exceeded.

Explanation / Answer

given equation

w/w0=l/l0

w=l/l0 * wo

=l/l0 sqrt 4 A/ pi

= -0.23 ( 9* 10^-4 sqrt 4 ( 0.30 * 10^-4 / pi

= 1.3 * 10^-6 m

(b)

F = A Y del l/ lo

= A Y ( 1/Poisson's ratio) del w/ wo

= pi ( 2 * 10^-2)^2 ( 2.1 * 10^11 Pa) ( 1/0.42) ( 0.10* 10^-3/ 2 * 10^-2)

= 3.1 * 10^6 N

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