I suggest you get a copy of Schaum's Outline Series, Engineering Mechanics,
McLean and Nelson. In
the Second Edition, Chapter 19, page 298, problem 24, describes a water jet.
The formulae are
there, just change the mass from that of water to air. Watch your units!
Unless you're planning on a pretty short test, I have my doubts about trying
to make a treadmill work--you're well in excess of the design load of the
machine, and the belt friction is only one area of concern.
There are SAE papers available that publish a "standard" design for test
machinery of this sort... a trip to a decent engineering library would be
time well spent.
Here is the problem from the both the second and fourth edition, chapter 20
Impulse and Momentum,
See several example problems dealing with this situation in the chapter.
Values are based on Standard Temperature and Pressure (STP)
water = 62.428 lb.cu ft
gravity = 32.174 ft/s/s
diameter of stream of water = 2 inches
velocity of water = 80 ft/sec
delts t = time interval of particles of water (Note: delta t cancels out, but
required in formula)
mass = (area of stream in sq ft) (velocity of stream in ft/s) ( density of water
in slug/(cu ft)
mass = (.25 x PI x (2/12) x (2/12)) x (80 x (delta t)) x (62.428 / 32.174)
mass = 3.3865(delta t) slugs Note: delta t cancels out below
Sum of the forces = mass(v" - v')
P(delta t) = 3.3865(delta t) x ( 0 - 80) = -270.92 lb
P = -271 lb
Force of water on the plate is +271 lb to the right.
The density of air at standard conditions (STP) is 0.074 lb/cu. ft. (Marks
Using the above with the density of air, the mass = 0.004(delta t) slugs and P 0.32 lb.
And 1000 lb / .32 lb = 3125 nozzles. You may modify your velocity and nozzle
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