A deep-sea diver is suspended beneath the surface of Loch...

A deep-sea diver is suspended beneath the surface of Loch Ness by a $100-\mathrm{m}$ -long cable that is attached to a boat on the surface (Fig. $\mathbf{P} 15.77$ ). The diver and his suit have a total mass of $120 \mathrm{~kg}$ and a volume of $0.0800 \mathrm{~m}^{3} .$ The cable has a diameter of $2.00 \mathrm{~cm}$ and a linear mass density of $\mu=1.10 \mathrm{~kg} / \mathrm{m} .$ The diver thinks he sees something moving in the murky depths and jerks the end of the cable back and forth to send transverse waves up the cable as a signal to his companions in the boat. (a) What is the tension in the cable at its lower end, where it is attached to the diver? Do not forget to include the buoyant force that the water (density $1000 \mathrm{~kg} / \mathrm{m}^{3}$ ) exerts on him. (b) Calculate the tension in the cable a distance $x$ above the diver. In your calculation, include the buoyant force on the cable. (c) The speed of transverse waves on the cable is given by $v=\sqrt{F / \mu}$ [Eq. (15.14)]. The speed therefore varies along the cable, since the tension is not constant. (This expression ignores the damping force that the water exerts on the moving cable.) Integrate to find the time required for the first signal to reach the surface.

Key Concepts

Integration over a Variable Medium

When dealing with a medium where parameters such as tension change along its length, integration is required to accurately compute quantities like the time a signal takes to travel along the medium. By considering the cumulative effect of varying conditions, integration allows one to sum up the contributions of small elements, leading to an overall solution for problems involving non-uniform systems.

Transverse Wave Propagation

The propagation of transverse waves along a string or cable is governed by a relationship between the wave speed, the tension in the medium, and its linear mass density, typically given by v = ?(F/?). This principle is essential in understanding how disturbances travel along a medium and how variations in tension along the cable can result in changes in wave speed.

Buoyant Force

Buoyant force is the upward force exerted by a fluid on an immersed object, and it is equal to the weight of the fluid displaced by the object. This concept, derived from Archimedes' principle, is crucial when analyzing forces on bodies in fluids because it can significantly reduce the effective weight of objects submerged in liquids, thereby affecting tensions, accelerations, and equilibrium conditions.

Tension Variation in a Cable

Tension in a cable that supports a weight distributed along its length is not constant but increases with depth due to the accumulation of the weight below any given point. This variation arises because each segment of the cable must support not only any attached load but also the weight of the cable underneath it, necessitating a detailed analysis to account for all forces acting along the length.

Linear Mass Density

Linear mass density refers to the mass per unit length of an object, such as a cable or rope. It is a fundamental concept in problems where the distributed mass along a length affects the overall behavior, such as determining the cable’s contribution to the tension or gravitational forces in systems that include cables, strings, or chains.

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