The Overloaded Lift Cable
A school's stage rigging uses a single steel cable to raise a 180 kg lighting bar. The cable's rated breaking stress is 5.0 × 10^8 Pa and its cross-sectional area is 2.4 mm². A technician replaces the bar with a heavier one of 260 kg and adds two 12 kg spotlights clamped at the ends. During a test lift the bar accelerates upward at 0.6 m/s². A second technician argues the cable is fine because the static weight is still below the breaking load, while a third notes the cable has been kinked and bent sharply at one point over the past year. The winch stalls briefly, producing a short jolt estimated at three times the steady acceleration.
1. Calculate whether the cable is safe under steady lifting, and explain how the upward acceleration changes the tension compared with the static case.
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Total mass = 260 + 2×12 = 284 kg. Static weight = 284 × 9.81 ≈ 2786 N. During steady upward acceleration the tension must both support the weight and provide the net force: T = m(g + a) = 284 × (9.81 + 0.6) ≈ 284 × 10.41 ≈ 2956 N. The breaking force is stress × area = 5.0 × 10^8 × 2.4 × 10^-6 = 1200 N. So the cable is already overloaded even statically (2786 N > 1200 N), and acceleration makes it worse. The second technician's reasoning is wrong on two counts: the static weight alone exceeds the breaking load, and the acceleration term adds roughly 6% on top. The correct judgement is that the lift must not proceed.
2. Assess the effect of the jolt and the kinked section on the safety margin, using the idea of stress concentration and fatigue.
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During the jolt the effective acceleration is about 3 × 0.6 = 1.8 m/s², giving T ≈ 284 × (9.81 + 1.8) ≈ 3298 N, nearly three times the rated breaking force. Even a brief excursion above the breaking stress can snap the cable, so the margin is negative, not merely small. The kink matters independently: a sharp bend concentrates stress locally, so the actual stress at the kink can exceed the nominal value calculated from force divided by area. Repeated bending over a year also causes fatigue, lowering the stress the cable can withstand. The judgement is that the cable should be replaced with a larger cross-section and the jolt source (winch stall) investigated before any further lifting.