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Drag Force Purpose: The purpose of this experiment is to determine the relationship between the drag force (due to air resistance) and velocity and shape. 1 Theory: The relationship between the drag force and velocity depends upon the mechanism involved. If viscous forces are causing the object to slow, we can use Stokes Law which tells us Fd(v) = 6rv 1 for a sphere of radius r moving at a velocity v where the viscosity of the fluid is . If inertial forces are causing the object to slow, for a sphere of radius r moving at a velocity v, 1 Fd(v) =2 CDAv2 2 where is the density of the medium and CD is a coefficient which depends on the form of the object and the medium. This equation comes from the fact that a falling object sweeps out a volume V = Avt where A is the cross sectional area, v is the velocity and t is the time interval. The number of particles the object collides with will be equal to N=V/m, where m is the mass of the particles. Each collision will produce an average change in momentum of p=mv. Use this information to derive equation 2. The equation of motion for an object of mass M falling in a fluid is dv M dt =M*g - Fd(v) 3 Where M* is the effective mass of the object taking into account the Archimedes Principle which results in a buoyant force due to the displaced fluid. (Luckily, M* is the value a scale measures when you place the balloon on it.) Note that if an object is allowed to reach terminal velocity vt Fd(vt) = M*g So the question remains...is Fd proportional to v or v2? Procedure: You will be given a sonic ranger, which is basically a motion detector, a balloon, and some masses to vary the mass of the balloon. Using a computer, you can use the sonic ranger to collect distance vs. time data for the balloon as it drops. You should be able to collect distance vs. time data for 5 to 7 different balloon masses (keeping the volume of the balloon constant). You should have between 5 and ten trials for each balloon mass. How will you determine whether Fd is proportional to v or v2? Determine the relationship between Fd and v and estimate CD. Using the differential equation 3, find v(t). This derivation is analogous to the one on page 62 of your text book (Classical Dynamics by Marion and Thornton.) 1 "Measuring air resistance in a computerized laboratory" by Ken Takanhash and D. Thompson, American Journal of Physics 67 (8) August 1999, p 709. Now we can determine how the drag force depends on the radius of the balloon. Essentially we repeat the experiment, only this time we change the radius of the balloon. Since the balloon reaches a terminal velocity, we have the following equation: Fd(vt) = M*g rm vn or M* A m = r vn g where K is some constant. From the previous experiment, you have determined the dependence of the drag force on v, whether n= 1 or 2. Now by making the appropriate graph, you can determine how the drag force depends on r. Does m = 1? 2? 3? Be careful to measure the mass of the balloon each time you change the radius. Even though you may not change the amount of material present, when you change the volume, you change the bouyant force acting on the balloon, which changes its effective mass, M*. F=bouyant force Weight = M*g = mg - Bouyant force Therefore, M*= (mg -Bouyant force)/g F = mg Theory: According to theory, whether an object falls with linear or quadratic drag depends on whether a certain parameter, the Reynolds number, is larger than a certain value. Look up what you can find on this quantity, and determine whether your findings are consistent with theory. 1 "Measuring air resistance in a computerized laboratory" by Ken Takanhash and D. Thompson, American Journal of Physics 67 (8) August 1999, p 709.