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 6.30 cm long, water washes away the sugar over a distance of 2.81 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.)
To find the greatest depth to which the bird dived, we can use the concept of pressure in fluids. The pressure at a certain depth in a fluid is given by the equation:
P = P₀ + ρgh
Where:
P₀ is the atmospheric pressure at the surface,
ρ is the density of the fluid,
g is the acceleration due to gravity,
and h is the depth or height of the fluid.
In this case, we know the length of the tube (6.30 cm) and the distance the sugar was washed away (2.81 cm). We can use these measurements to find the depth to which the bird dived.
First, we need to convert the lengths to meters for consistency. 6.30 cm converted to meters is 0.063 m, and 2.81 cm converted to meters is 0.0281 m.
Next, we can use the concept of pressure equilibrium to determine the pressure at the open end of the tube (P_open) and the pressure at the depth to which the bird dived (P_depth). Since the air in the tube is at constant temperature, we can assume that the pressure at the open end of the tube is equal to the atmospheric pressure (P₀).
Therefore, we have:
P_open = P₀
P_depth = P₀ + ρgh
Now we can equate the pressures and solve for h (depth):
P_open = P_depth
P₀ = P₀ + ρgh
Canceling out P₀, we get:
0 = ρgh
Now we can substitute the values into the equation:
0 = ρgh
0 = (density of ocean water) * g * h
Since we are looking for the greatest depth, we can assume that the bird dived to the maximum depth, which means the pressure at the open end of the tube (P_open) is equal to the pressure at the greatest depth (P_depth).
Therefore, we have:
P_open = P_depth
P₀ = P₀ + ρgh
0 = ρgh
Now, we can plug in the known values:
0 = (1030 kg/m³) * (9.8 m/s²) * h
Simplifying the equation, we get:
0 = 10094 h
Since h represents the depth to which the bird dived, we can solve for h:
h = 0 / 10094
h = 0
Therefore, the greatest depth to which the bird dived is 0 meters. It means that the bird did not dive at all, as indicated by the sugar being washed away at a distance of 2.81 cm from the open end of the tube.