4.2.2.P4
Typically when we describe falling objects in a physics class, we say “ignore resistive forces”.
In this problem, we’ll estimate which resistive force dominates for different objects. If we have a sphere moving in a fluid of density, d_{fluid}, and the object has a radius, R, then it will experience two resistive forces, drag and viscosity as given by the equations:
For air and water, here are the values of the parameters that occur in these equations.

Density (d)

Drag coefficient (C)

Viscosity (μ)

Air

1.2 kg/m^{3}

0.20

2.0 x 10^{5} Ns/m^{2}

Water

1.0 x10^{3} kg/m^{3}

0.25

0.8 x 10^{3} Ns/m^{2}

Since we are only interested in the approximate scale of the forces, we will model our objects by spheres,
even though they are not really spherical. (Corrections can be expected to be less than a factor of 10.)
A. Consider the larva of a daphnia, a small crustacean, shown at the right. It lives in water and at certain times in its lifecycle, it goes dormant and sinks passively to the bottom. Model it as a sphere of radius = 0.3 mm, density = 1.3 g/cm^{3}, and mass = 0.3 mg. Typical observed passive sinking velocities are on the order of 2 mm/s. Draw a free body diagram showing all the forces acting on the daphnia larva as it sinks, being careful to label each of the forces. Calculate the ratio of the drag to the viscous force,
R = F_{drag} / F_{viscous}
and tell which force is more important for the passive sinking of the larva.

Source: Wikimedia commons 
B. As our second example, let’s consider a northern flying squirrel, shown in the figure at the right. For this problem, we will model it as a sphere (!) of radius = 15 cm and mass = 200 g. Typical observed velocities are on the order of 50 cm/s. In the space below calculate the ratio of the drag to the viscous force,
R = F_{drag} / F_{viscous}
and mark which force is more important for the falling of the squirrel. Show your work.


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