A deep-sea diver is suspended beneath the surface of Loch Ness by a cable of len

ID: 1613760 • Letter: A

Question

A deep-sea diver is suspended beneath the surface of Loch Ness by a cable of length h = 100 m that is attached to a boat on the surface (Figure 1) . The diver and his suit have a total mass of m = 120 kg and a volume of V = 8.40×102 m3 . The cable has a diameter of 2.10 cm and a linear mass density of = 1.15 kg/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.

Part 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 water = 1000 kg/m3 ) exerts on him.

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Part B

Calculate the tension in the cable a distance x above the diver. The buoyant force on the cable must be included in your calculation.

Express your answer in terms of the variables , water, d, x, m, V, and appropriate constants.

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Part C

The speed of transverse waves on the cable is given by v=F/. The speed therefore varies along the cable, since the tension is not constant. (This expression neglects 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.

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T = N 100 m m 120 kg

Explanation / Answer

part a )

T = mg - Fb

FB = buoyant force = rho * V *g

rho =dater density = 1000 kg/m^3

m = 120 kg

V = 8.4 x 10^-2 m^3

T = 352.8 N

part b )

The increase in tension will be the weight of the cable between the diver and the point at x, minus the buoyant force. This increase in tension is then

mu*x*g - rho*A*x*g

A = area = pi*r^2

r = 2.1/2 = 1.05 x 10^-2 m

= 7.876 N/m

tension as a function of x is

F(x) = 352.8 + (7.876)*x

part c )

F(x) = Fo + a*x

Fo = 352.8 N

a = 7.876 N/m

speed of transverse waves as a function of x is v = dx/dt = sqrt([Fo+ax]/mu)

integral dt = integral sqrt(mu) / sqrt(Fo+ax) * dx from 0 to L

t = 2*sqrt(u)/a * [ sqrt(Fo+a*L) - sqrt(Fo) ]

t = 4.08 s

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A deep-sea diver is suspended beneath the surface of Loch Ness by a cable of len

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