An 80.0 kg hiker is trapped on a mountain ledge following a storm. A helicoptar rescuse the hiker by hovering above him and lowering a cable to him. The mass of the cable is 8.00 kg, and its length is 15.0 m. A sling of mass 70.0 kg is attached to the end of the cable. the hiker attaches himself to the sling, and the helicopter then accelerates upward. terrified by hanging from the cable in midair, the hiker tries to singnal the pilot by sending transverse pulses up the cable. a pulse takes 0.250 s to travel the length of the cable. what is the acceleration of the helicopter?

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Imagine the effected of the acceleration of the helicopter on the cable. The greater the upward acceleration, the larger the tension in the cable. In turn upward acceleration, the larger the tention in the cable. In turn, the larger the tension, the higer the speed of pulses on the cable.
This problem is a combination of one involving the speed of pulse on a string and one in which the hiker and sling are modelled as a particle under a net force.
Use the time interval for the pulse to travel from the hiker to the helicopter to find the speed of the pulse on the cable.
$v = \frac{Δ x}{Δ t} = \frac{15.0 m}{0.25 s} = 60.0 m / s$
The tension in the cable: using $v = \sqrt{T l μ}$
Model the hiker and sling as a particle under net force, nothing that the acceleration of this particle of mass m is the same as the acceleration of the helicopter:
$\sum F = m a \Rightarrow T - m g = m a$
solved for the acceleration
$a = \frac{T}{m} - g = \frac{μ v^{2}}{m} - g = \frac{m_{c a b \leq} v^{2}}{l_{c a b \leq} m} - g$ substitute numerical values:
$a = (8.00 k g) {(60.0 m / s)}^{2} \frac{)}{(15.0 m) (150.0 k g)} - 9.80 m / s^{2} = 3.00 m / s^{2}$
A rela cable has stiffness in addition to tension, stiffness tends to return a wire to its original straight-line shape even when it is not under tention . for example, a piano wire straightnes if released from a curved shape, package-wrapping string does not.
Stiffness represents a restoring force in addition to tension and increase the wave speed. consequently, for a real cable ,the speed of $60.0 m / s$ that we derermined is most likely associated with a smaller acceleration of the helicopter.

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Updated on:21/07/2023

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