What is the wind speed in m/s required to keep a great white shark suspended in midair during a summer day on the California beach?
Details and assumptions
Model the shark as a horizontal cylinder 6 m long and 1 m in diameter.
The air is at a pressure of 1 atm and a temperature of 30∘C.
The density of the shark is that of water, 1000 kg/m^3.
Assume that the wind gust that keeps the shark suspended is blowing straight upwards, and that the air molecules bounce off the shark elastically.
The acceleration of gravity is −9.8 m/s
^2.
ma, the molar mass of air, is 29 g/mol.
To calculate the wind speed required to keep the great white shark suspended in midair, we can use the principle of buoyancy.
1. Determine the weight of the shark:
The weight of an object can be calculated using the formula: weight = mass x gravity, where the mass is the density of the shark multiplied by its volume.
Given that the density of the shark is 1000 kg/m^3 and the volume can be calculated as the product of the length (6m) and the cross-sectional area (πr^2) where r is the radius (0.5m):
Mass = density x volume = 1000 kg/m^3 x (6m x π x (0.5m)^2)
2. Calculate the pressure difference:
The pressure difference between the top and bottom of the shark is caused by the upward force of the wind. Assume the pressure at the bottom of the shark is Patm (1 atm) and pressure at the top is Patm + ΔP. The pressure difference can be calculated as ΔP = weight / (area of the shark) = weight / (π x (0.5m)^2)
3. Calculate the average speed of the air molecules:
The average speed of the air molecules can be calculated using the root mean square speed formula: v = √(3kT / m), where k is the Boltzmann constant, T is the temperature in Kelvin, and m is the molar mass of air.
First, we convert the given temperature from Celsius to Kelvin: T = 30°C + 273.15K.
Then calculate the average speed of air molecules using the formula.
4. Determine the wind speed:
The wind speed is equal to the average speed of the air molecules when they collide with the shark elastically.
Plug in all the given values into the formulas, and perform the necessary calculations to find the wind speed required to keep the great white shark suspended in midair on a summer day on the California beach.