1. A pendulum has an oscillation period T which is assumed to depend upon its length L, bob mass m, angle of swing ??, and the acceleration of gravity. A pendulum 1 m long, with a bob mass of 200 g, is tested on earth and found to have a period of 2.04 s when swing at an angle of 20 degree. A similarly constructed pendulum, with L=30 cm and m=100 g, is to swing on the moon (g=1.62 m/s2) at an angle 20 degree. What will be its period?
2. A prototype water pump has an impeller diameter of 2 ft and is designed to pump 12 ft3/s at 750 rpm. A 1-ft diameter model pump is tested in 20oC air at 1800 rpm, and Reynolds number effects are found to be negligible. For similar conditions, what will the volume flow of the model in ft3/s? If the model pump requires 0.082 hp to drive it, what horsepower is required for the prototype?
3. Your ME6666 Professor asks you to experimentally identify the parameters that determine the terminal velocity, vT, of spinning, smooth, aluminum spheres that are in free fall in different fluids. You suspect the possible parameters that affect the terminal velocity are: the initial spin rate ω, the fluid density ρ, the fluid dynamic viscosity μ, and the sphere density diameter D. Since the spheres are made of the same material, so the sphere density is not a parameter. You collect measurements for different sized material, different fluids and different spin rates, as shown below.
|
vT (m/s)
|
ω (1/s)
|
ρ (kg/m3)
|
μ (kg/(m.s))
|
D (m)
|
|
0.5
|
1500
|
1200
|
0.001
|
0.25
|
|
0.62
|
500
|
500
|
0.0002
|
0.1
|
|
4.22
|
2000
|
600
|
0.005
|
0.3
|
|
9.9
|
750
|
5
|
0.00005
|
0.15
|
|
64.7
|
1400
|
1.4
|
0.00003
|
0.05
|
After you have collected these data, one of your friends suggests that, in fact, the fluid viscosity isn't important; then another friend suggests that the spin rate isn't important. You decide to approach the problem using the dimensional analysis taught by the professor.
a. Determine the non-dimensional parameters that govern the problem if all of the parameters are important. Use ??, ?? and D as repeating parameters.
b. Determine the non-dimensional parameters that govern the problem if the fluid viscosity isn't important;
c. Determine the non-dimensional parameters that govern the problem if the spin rate isn't important.
d. Based on your experimental data, which of the parameters seems irrelevant? Explain.