GRADES 9–12
NGSS: Motion and Stability: Forces and Interactions:
Analyze data to support the claim that Newton’s second law of motion describes the mathematical relationship among the net force on a macroscopic object, its mass, and its acceleration.
When gravitational forces are not important, other forces begin to dominate the flow of water. All of these forces push on the water and cause it to move; some push harder than others. The relative influence of the different types of forces are summarized
in a series of dimensionless numbers. For example, the Froude number is the ratio of the gravitational force on a bit of water and its inertia. The formula for the Froude number is
where
U_{0 }is a typical speed of the flowing water
g_{0 }is the gravitational (or other bulk) acceleration
L_{0 }is a typical length of the waterflow
In a pitcher of water on the surface of the Earth, the typical speed will be quite small, maybe a tenth of a meter per second; the gravitational acceleration will be about ten meters per second squared; and the typical length will be about a tenth of a meter, giving a Froude number of about 0.1. The number is dimensionless; if you express
the quantities in English units (U_{0}~ 0.32 ft/sec, g_{0} = 32.2 ft/sec^{2}, L_{0} ~ 0.32 ft) you will get the same result of Fr~ 0.1. Small Froude numbers indicate that the gravitational force is important in a flow; large Froude numbers
indicate that gravity is not important.
Another dimensionless number is the Eötvös (pronounced “Ootvoos”—the “ö” sounds like the “oo” in “foot”) number, also called the Bond number. This number describes the relative importance of gravity and surface tension in a flow; its formula is
where
Δ_{ρ }is the density of the water (strictly speaking the difference in density of the water from the surrounding air)
g_{0 }is the gravitational (or other bulk) acceleration
L_{0 }is a typical length of the flowing water
σ is the surface tension of the water
If the Eötvös number
is much larger than one, then gravity predominates over surface tension. If it is much less than one, though, surface
tension predominates over gravity.
The surface tension of water
in air is about
0.07 Kg/s^{2}; the density of water is about 1000 Kg/m^{3}. On the surface of the Earth, where the
gravitational acceleration is 9.8 m/s^{2}, a water droplet 0.001 meters
(one millimeter) across (just less than a quarter of an inch) will have an
Eötvös number of 1000 Kg/m^{3} × 9.8 m/s^{2} × ( 0.001 m )^{2}
/ 0.07 Kg/s^{2} = 0.14, meaning that surface tension will pull the
droplet into a spherical shape even though gravity is pulling it out flat. In the Space
Station video in which astronaut Chris Hadfield wrung out a washcloth about
0.2 meters long in the presence of an exceptionallysmall gravitational
acceleration (let us use one microgravity, or about 0.00001 m/s^{2},
for convenience, although the true number could be smaller by a factor of ten),
the Eötvös number is about 0.006. In
this case, surface tension is very much more important than gravity in
determining how the water flows around Mr. Hadfield’s washcloth.
There are
many more dimensionless groups from various fields of science and engineering. Many of them are very specialized; others,
like the Mach number, are known by the general public.
Sixty Years Ago in the Space Race:
December 6: The first American attempt
to launch a satellite into orbit failed spectacularly when the Vanguard TV3
exploded a few seconds after liftoff in an attempt to launch a satellite into
orbit.
