Show that the variation of atmospheric pressure with altitude is given by, where a is atmospheric pressure at some reference level, and role="math" localid="1663776653925" is the atmospheric density at this level. Assume the decrease in atmospheric pressure over an infinitesimal change in altitude (so that the density is approximately uniform over the infinitesimal change) can be expressed from Equation 14.4 asrole="math" localid="1663776670018" . Also assume the density of air is proportional to the pressure, which, as we will see in Chapter 20, is equivalent to assuming the temperature of the air is the same at all altitudes.
It is proved that the variation of atmospheric pressure with altitude is given by .
Pascal's law states that when pressure is applied to a confined fluid, the pressure is transmitted undiminished to every point in the fluid and to every point on the walls of the container.
The pressure in a fluid at rest varies with depth h in the fluid according to the expression
Where, is the absolute pressure, is the atmospheric pressure and is the density of the fluid, assumed uniform, is the gravitational acceleration constant, and is the height.
The incremental version of is .
Assume that the density of air is proportional to pressure, or .
Combining these two equations you have,
Integrating both the above sides,
Defining then gives .
A spherical vessel used for deep-sea exploration has a radius of and a mass of role="math" localid="1663750322684" . To dive, the vessel takes on mass in the form of seawater. Determine the mass the vessel must take on if it is to descend at a constant speed of role="math" localid="1663750448075" , when the resistive force on it is in the upward direction. The density of seawater is equal to .
On October 21, 2001, Ian Ash pole of the United Kingdom achieved a record altitude of powered by toy balloons filled with helium. Each filled balloon had a radius of about and an estimated mass of . (a) Estimate the total buoyant force on the balloons. (b) Estimate the net upward force on all balloons. (c) Ash pole parachuted to the Earth after the balloons began to burst at the high altitude and the buoyant force decreased. Why did the balloons burst?
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