PHYS 2425 – Engineering Physics I

Air Resistance

 

Leader: _____________________________  Recorder: ___________________________

Skeptic: _____________________________ Encourager: _________________________

 

Materials

Laptop

LabPro

Motion Detector

Coffee Filters (small basket type)

Ring stand with right angle and jaw clamp

 

Introduction

      In this lab we will investigate the effect that resistive forces have on the motion of a falling object.  A resistive force such as air resistance depends on the velocity of the object.  The simplest model for a velocity dependent resistive force is that the force is proportional to the velocity,  F(v) = -bv.  The constant b depends on the properties of the medium through which the object drops as well as the shape and geometry of the object.  This model tends to work well for small objects moving slowly.  A second and usually more realistic model commonly used to model air resistance is the air resistance is quadratic in the velocity.  We typically write the resistive force for the case of a quadratic model as F(v) = - 1/2 CrAv2, where  r is the density of the medium through which the object falls, A is the cross sectional area of the object, and C is a constant known as the drag coefficient.

 

Q1.  Draw a free body diagram for a falling object including air resistance. 

 

 

 

 

Q2.  Write Newton’s second law for the free body diagram that you have drawn.

 

 

We will now analyze the behavior of the equation qualitatively.  If we drop the object from rest, it initially has a velocity of zero, and consequently the air resistance is zero. 

 

Q3.  How does the magnitude of the resistance force change as the object falls?

 

 

 

Q4.  If the magnitude of the resistance force increases as the object falls, then what happens to the net force on the object as it falls?

 

 

Q5.  If the net force becomes 0, the velocity becomes constant.  This velocity is called the terminal velocity.  Find an expression for the terminal velocity for the case of a linear resistive force in terms of b, g, and m.

 

 

 

Q6.  Find an expression for the terminal velocity for the case of a quadratic resistive force in terms of c, ρ, g, A, and m.

 

 

Q7.  Compare the two expressions for the terminal you have found.  We can differentiate them by how they depend on mass.  Describe how each expression for the terminal velocity depends on the mass.

 

 

 

Procedure

      We will employ the motion detector to obtain data for this lab.  The experimental set up is shown in figure 1. 

 

Figure 1 Schematic of experimental set up

1.  Set Up

      Clamp the motion detector so that it faces down, is level, and is as high as possible above the floor.  Connect the motion detector to DIG/SONIC 1 of the LabPro.

 

      We will use as the objects to drop in this experiment stacks of various numbers of coffee filters.  These make ideal objects for this experiment.  The mass is small making air resistance important.  Also the mass can be easily adjusted by simply adding more coffee filters to the stack while maintaining an effectively constant cross sectional area.         Start up LoggerPro and open the experiment file titled “Motion Detector” by following the path “Probes and Sensors”=>”Motion Detector”=>”Motion Detector”.

 

2.  Data Acquisition

      Make a stack of 2 coffee filters.  Use the electronic scale to determine the mass of the stack of coffee filters.

      Hold the stack of 2 coffee filters away from your body at table height directly below the motion detector with the flat end down at least .5 m below the motion detector. Click on the Collect Button.  Once the motion detector starts clicking steadily, drop the filters.

      We will now extract the relevant data from our graph.  Figure 2 shows qualitatively what a graph of distance vs. time might look like.  Initially the distance grows quadratically in time, but as the terminal velocity is approached, the distance tends to grow linearly in time.  The slope of the linear portion of the distance versus time graph is the terminal velocity.  We wish to extract this slope and thus obtain the terminal velocity.

 

Figure 2  Schematic of  “Typical” Data

      Click on the Position vs. time graph so that it is active.  Click and drag over the region of the data that appears to be linear.  Click on the linear fit button (looks like a tangent line with the letter R on it).  This will automatically give you the best fit line and put the equation of the best fit line in a tiny box on the screen.  Record the mass of the stack of filters and the terminal velocity in the data table below.

 

Figure 3  Data Table

Trial

Mass (kg)

Terminal Velocity

(m/s)

1

 

 

2

 

 

3

 

 

4

 

 

5

 

 

 

Q8.  Sketch a graph of velocity vs. time in the space below. 

 

 

 

 

 

Q9.  Indicate on the graph the region where terminal velocity was reached.  What feature of the graph tells you that terminal velocity was reached?

 

 

Q10.  Sketch a graph of acceleration vs. time in the space below.

 

 

 

 

Q11.  Indicate on the graph the region where terminal velocity was reached.  What feature of the graph tells you that terminal velocity was reached?

 

 

3.  Repeat the Experiment for Different Masses

      Repeat your series of measurements four more times, each time increasing the number of coffee filters by 1.  Each trial, record the mass of the coffee filters and the terminal velocity you obtained in the data table.

 

Data Analysis

      The information we wish to extract from our data is which of the models for air resistance better describes the data.  The linear model for air resistance gives a result where the terminal velocity is a linear function of the mass, and the quadratic model yields a terminal velocity which is proportional to the square root of the mass.  In either case, we expect a result of the form v = kmn (1) where k is some constant and n is some power.  We wish to determine the exponent.

 

Q13.  List the value of the exponent for the linear case

n =

 

Q14.  List the value of the exponent for the quadratic case

n =

 

      A common data analysis tool to extract information about exponents is to plot a log-log plot. In a log-log plot instead of plotting y vs. x we plot log(y) vs. log(x).  The log can be to any base, but typically base 10 is chosen.  If we consider v = kmn (1),  let us examine what happens when we take the log of both sides.  Remember the important properties of logarithms i) log(AB) = log(A) + log(B) and ii)  log(An) = n log(A). 

 

Q15.  Take the log of both sides of (1) and use the properties of the logarithms to obtain an expression for log(v) in terms of log(m), n and log(k)

 

 

 

Q16.  Based on your answer to Q15 what type of graph should you obtain if you graph log(v) vs. log(m).

 

 

 

Q17.  What will be the slope of the graph of log(v) vs. log(m) if the linear air resistance model is a better description of this experiment?

 

 

 Q18.  What will be the slope of the graph of log(v) vs. log(m) if the quadratic air resistance model is a better description of this experiment?

 

 

Use LoggerPro or Excel to graph log(v) vs. log(m) and perform a linear regression analysis to determine the slope. 

 

Q19.  Record your slope

 

 

Q20.  Is your slope consistent with either of the models discussed in this lab?  Explain your answer.

 

 

Q21.  Attach a copy of your log-log plot to the handout.

 

 

Further Analysis

      In this part we will find an analytical expression for the position as a function of time for the case of linear air resistance.  In Q2 we should have written Newton's 2nd law for the falling mass a something like bv – mg = -ma where we have chosen up as the positive direction.  Since the force is velocity dependent, it is helpful to write a as dv/dt.  Doing so gives the equation for the motion as (2)  .  Newton's second law in this case gives a differential equation for the velocity.  Consequently, the equation obtained from Newton's 2nd law is often referred to as the equation of motion.  The equation can be solved using a technique called separation of variables.  When a differential equation is separable, it can be rearranged so that it looks like dv/dt = f(v) g(t). 

 

Q22.  To see this, divide both sides of (2) by (mg – bv)m.

 

 

 

Q23.  The next step in solving by separation of variables is multiplying both sides by dt.  Carry this out.

 

 

 

Q24.  Part of the term on the left should look like dv/dt dt.  What does this equal?  Replace it in the answer to Q23.

 

 

Q25.  Now integrate both sides of  Q24.  Don't forget that you will need an arbitrary constant.  Hint: 

 

 

 

 

 

 

Q26.  Solve the expression you obtained in Q25 for v.  Evaluate the constant by using that v(0) = 0.

 

 

 

Q27.  You should notice that your answer involves an exponential function.  The terminal velocity can be found by considering the limit at t →∞.  Take .  Do you get the same result as you found in Q5?