To measure how far below the ocean surface a bird dives to catch a fish, a scien

Question

To measure how far below the ocean surface a bird dives to catch a fish, a scientist uses a method originated by Lord Kelvin. He dusts the interiors of plastic tubes with powdered sugar and then seals one end of each tube. He captures the bird at nighttime in its nest and attaches a tube to its back. He then catches the same bird the next night and removes the tube. In one trial, using a tube 7.00 cm long, water washes away the sugar over a distance of 3.21 cm from the open end of the tube. Find the greatest depth to which the bird dived, assuming the air in the tube stayed at constant temperature. (Assume the density of the ocean water is 1030 kg/m3.)

Explanation / Answer

Air compress isothermally, hence P1 * V1  = P2 * V2

P1 * A * L = P2 * A * L1

where initial pressure P1   = 1.01 * 105  N/m2   (atmospheric pressure)

A = area of cross section of tube

L = total length of tube = 6.50 cm

L1   = Length of tube filled with air only

= 6.50 - 2.70

= 3.80 cm

1.01 * 105 * 6.50 = P2 * 3.80

Air pressure inside tube P2   = 1.01 * 105 * 6.50 / 3.80

= 1.73 * 105   N/m2

At depth H, this pressure P2 plus pressure of water inside the tube must balance the pressure of water

P + P1   = P2 + Pwater

* H * g + 1.01 * 105   = 1.73 * 105   + * (L - L1) * g

here (L - L1   = 2.70 cm = 0.0270 m) is length to which water washes away sugar coating

1059 * H * 9.8 = (1.73 * 105 - 1.01 * 105) + 1059 * 0.0270 * 9.8

= 7.23 * 104

Depth of dive H = 7.23 * 104 / 1.04 * 104

= 6.95 m

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