A high-speed lifting mechanism supports a 880-kg object with a steel cable that

Question

A high-speed lifting mechanism supports a 880-kg object with a steel cable that is 39.0 m long and 4.00 cm2 in cross-sectional area.

(a) Determine the elongation of the cable.
__________ mm

(b) By what additional amount does the cable increase in length if the object is accelerated upwards at a rate of 2.7 m/s2?
______________ mm

(c) What is the greatest mass that can be accelerated upward at 2.7 m/s2 if the stress in the cable is not to exceed the elastic limit of the cable, which is 2.2 ? 108 Pa?
_____________ kg

Explanation / Answer

E = F / A / ? L /L
E ( ? L /L ) = F/ A
? L = F * L / A * E
E = young modulus steel 200 10^9 N/m2
F = m * g = 880 * 9.8 = 8624 N
L = 39m
A = 4 * 10^-4 m2
? L = F * L / A * E = 8624 * 39 / 4 *10^-4 * 200 * 10^9 = 4.20 * 10^-3
? L = 4.20 mm

(b)F1 = m (9.8 + 2.7 ) = 880 * 12.5 = 11000 N
? L1 = F1/F * ? L = 1.2755 * =5.357 mm

(c)Fmax = A stress max = 4* 10^-4 * 2.2 10^8 = 8.8 * 10^4 N

mass max = F / ( g + 2.7 ) = 88 10^3 / 12.5 = 7.04 10^3 = 7040kg

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