Problem 9.17 Feather in tornado.
In this project you will learn to use Newton’s laws and the force model for air resistance in a wind field to address the motion of a light object in strong winds. We start from a simple model without wind and gradually add complexity to the model, until we finally address the motion in a tornado. Motion without wind:
First, we address the motion of the feather without wind.
(a) Identify the forces acting on a feather while it is falling and draw a free-body diagram for the feather.
(b) Introduce quantitative force models for the forces, and find an expression for the acceleration of the feather. You may assume a quadratic law for air resistance.
(c) If you release the feather from rest, its velocity will tend asymptotically toward the terminal velocity, \(v_T\) . Show that the terminal velocity is \(v_T\) = \(−(mg/D)^{1/2}\), where D is the constant in the air resistance model.
(d) We release the feather from a distance h above the floor and measure the time t until the feather hits the floor. You may assume that the feather falls with a constant velocity equal to the terminal velocity. Show how you can determine D/mg by measuring the time t. Estimate D/mg when you release the feather from a height of 2.4 m above the floor and it takes 4.8 s until it hits the floor.
(e) We will now develop a more precise model where we do not assume that the velocity is constant. You release the feather from the height h at the time t = 0 s. Find the equation you have to solve to find the position of the feather as a function of time. What are the initial conditions?
(f) Write a program that solves this equation to find the velocity and position as a function of time t. Use the parameters you determined above, and test the program by ensuring that it produces the correct terminal velocity.
(g) Fig. 9.19 shows the position and velocity calculated with the program using the parameters found above. Was the approximation in part (d) reasonable? Explain your answer. Model with wind: We have now found a model that can be used to find the motion of the feather. We will now find the motion of the feather in three dimensions while it is blowing. The velocity of the wind varies in space, so that the wind velocity w is a function of the position r. We write this as w = w(r).
(h) Find an expression for the acceleration of the feather. The expression may include the wind velocity w(r). Let the z-axis correspond to the vertical direction.
(i) Assume that the feather is moving in an approximately horizontal plane—that is you may assume that the vertical acceleration is negligible. How does the wind have to blow in order for the feather to move in a circular orbit of radiusr0 with a constant speed v0?
Motion in a tornado: For a tornado with a center at the origin, the wind velocity is expected to be approximately given by the model:
\[\boldsymbol{w}(\boldsymbol{r}) = u_0\boldsymbol{r}e^{−r/R}\hat{u_θ} = u_0(-y, x,0)e^{−r/R}\hat{u_θ} , \space \space (9.79) \]