When a maple leaf seed of mass m falls from a tree, the air causes the seed to rotate with angular frequency ω, which generates an upward lift force. Since the direction of this force is also opposite to the falling motion of the seed, it is equivalent to the drag force FD, hence the phrase drag-based lift that many birds and insects take advantage of to fly. By increasing the amount of time it takes to land, the seed can be carried further away by the wind. The shape of the maple seed is irregular, so to account for the distribution of the cross-sectional area, the radius of gyration R2 (based on the second moment of area) lets us treat the maple seed as a regular shape. As the seed falls, the air with density rho and viscosity µ will eventually balance the acceleration due to gravity g and cause the seed to fall at an average terminal velocity Vt. [p] = N, [M] = M[Vt] = > [R2] = L, [g] = , (FD] = ML, [w] = 1, [m] = M 1.
1. Using air density, radius of gyration and angular frequency, determine the II-groups for: viscosity II,, drag force IId, and terminal velocity II).
2. These II-groups are sometimes referred to as the dimensionless form of the parameter. Determine the non-dimensional mass m* = Ilm, and gravity g* = II.
3. Show that when the dimensional drag force is equal to the dimensional weight, the dimensionless drag force is equal to the product of dimensionless mass and dimensionless gravity.
4. An undergraduate researcher wants to measure the relation between the weight and terminal speed of a maple seed. A 4-scale model is developed and held submerged in a water channel with controllable velocity. The model has a bearing that allows it to spin freely without friction.
A) What is the ratio of the angular frequency of the model to that of the real maple seed?
B) If the real maple seed falls at a steady-state speed V =1", what speed should the water channel velocity be set?
C) The student measures a force in the water channel F = 5.1 mN. What is the corresponding mass (in mg) of the real maple seed?