Objective:

In this lab you will track a helium-filled balloon as it rises in air.  The balloon is released at sea level in a standard atmosphere.  It is subject to the buoyant force, the gravitational force, and the drag force.

Open a Microsoft Word document to keep a log of your experimental procedures and results.  Complete all the tasks (in blue).  Answer all questions.

Exercise 1

Use an on-line simulation from the University of Colorado PhET group to experiment with a helium balloon, a hot air balloon, or a rigid sphere filled with different gases and discover what makes some balloons float and others sink.
Link to the simulation:  Balloons and Buoyancy

Explore the interface!

The chamber
You can pump two types of molecules into the chamber.
You can specify constant volume, pressure, or temperature.
You can work in an environment with or without gravity.
You can heat up the gas in the chamber.
You can use various measuring tools.

The hot air balloon
The hot air balloon in the simulation is open to the atmosphere at the bottom.  The pressure inside the balloon is always just slightly above atmospheric pressure, to keep the balloon inflated.
Describe what you can do to in the simulation to make the balloon rise and then float when the chamber pressure is ~1 atm and the temperature is ~300 K..  When the balloon is floating, comment on the species and compare temperature and the density of the molecules inside and outside of the balloon.

The rigid hollow sphere
The rigid hollow sphere is closed to the atmosphere and cannot expand or contract.
Describe what you can do to in the simulation to make the sphere rise and then float when the chamber pressure is ~1 atm and the temperature is ~300 K..  When the sphere is floating, comment on the species and compare temperature and the density of the molecules inside and outside of the sphere.

The Helium balloon
The Helium balloon in the simulation is closed to the atmosphere but can expand or contract.
Describe what you can do to in the simulation to make the balloon rise and then float when the chamber pressure is ~1 atm and the temperature is ~300 K..  When the balloon is floating, comment on the species and compare temperature and the density of the molecules inside and outside of the balloon.

Exercise 2

Assume that at sea level the temperature is T0 = 15 oC = 288 K, the pressure is P0 = 1 atm = 10131 Pa, and the density of the air is ρ0 = 1.225 kg/m3.  Assume that the Temperature T varies with altitude y as T = T0 - ay, a = 0.0065 K/m.  Then the pressure varies with altitude as (P/P0)0.19 = ( 1 - ay/T0), or P/P0 = [(T0 - ay)/T0]5.255.  The density ρ of the air is proportional to the pressure divided by the temperature (according to the ideal gas law).  We therefore have ρ/ρ0 = PT0/(P0T) = [(T0 - ay)/T0]4.255 = [(1 - 0.0002257y)]4.255.  Given ρ0 and T0  the density of the air is only a function of the altitude y.

Assume a foil balloon with radius r = 15 cm is filled with He.  The pressure in the balloon is 1.1 atm.  The mass of the balloon, including the He is 10 g.  The balloon is released at sea level.  It is acted on by a buoyant force, which is equal to the weight of the displaced air, by the gravitational force mg, and, once it is moving, by a drag force.

The buoyant force is ρVg j, where V = 0.014 m3 is the volume of the balloon.  The buoyant force changes with altitude.  The weight of the balloon is -mg j = -0.1176 j.  It is constant if we neglect small variations of g with altitude.  The drag force has magnitude R = ½DρAv2 = 0.0177ρv2 if we use D = 0.5 and A = πr2.  Its direction is opposite to the direction of the velocity v.  The total force acting on the balloon is F = ρVg j - 0.1176 j - 0.0177ρvv.  Its acceleration is a = F/m.

You will first explore the motion of the balloon in a 1 s time interval after it has been released at t = 0.  At t = 0, its velocity is zero, its position is zero, the density of the air is ρ = ρ0 = 1.225 kg/m3, and the acceleration of the balloon is F/m = (ρVg - 0.1176) j/0.01 = (0.1372ρ - 0.099)/0.01 j in SI units.  In a short time interval Δt = 0.001 the altitude y changes Δy = ½aΔt2, the velocity changes by Δv = aΔt and the density changes to ρ = 1.225(1 - 0.00002257y)4.255.

Prepare sheet 1 of an Excel workbook as shown in the figure below.  All numbers in the spreadsheet represent physical quantities in SI units.
 A B C D E 1 t y v density a 2
On the menu bar click the developer tab and choose macro. ( If the developer tab is not available, click the File tab, Options, Customize Ribbon, and select the Developer check box.)  Choose a macro name, for example "balloon1", and click create.
A Visual Basic for Application (VBA) window will open.  It will contain two entries.
 Sub balloon1() End Sub
Enter the following program.  (You can copy and paste.)
 Sub balloon1() Set w = Worksheets("sheet1") t = 0 v = 0 y = 0 density = 1.225 a = (density * 0.1372 - 0.098) / 0.01 i = 50 j = 2 Do While t < 1     If i = 50 Then         a1\$ = "a" & j         b1\$ = "b" & j         c1\$ = "c" & j         d1\$ = "d" & j         e1\$ = "e" & j         w.Range(a1\$).Value = t         w.Range(b1\$).Value = y         w.Range(c1\$).Value = v         w.Range(d1\$).Value = density         w.Range(e1\$).Value = a         i = 0         j = j + 1     End If     t = t + 0.001     y = y + v * 0.001 + 0.5 * a * 0.00001     v = v + a * 0.001     density = 1.225 * (1 - 0.00002257 * y) ^ 4.255     a = (density * 0.1327 - 0.098 - 0.0177 * density * v * Abs(v)) / 0.01     i = i + 1 Loop End Sub
This program increments the time t in 0.001s intervals and calculates the position y, the velocity v, the density r, and the acceleration as a function of time using the kinematic equations and the equation for the density as a function of altitude.  Every 0.05 s the program writes the results of the calculations into a new row of the spreadsheet.  The program stops when t = 1 s.
 Close the VBA window. Click on an empty cell in your worksheet.  On the menu bar click Macro and run your program.  Watch how the program creates the spreadsheet.
Now produce graphs of y, v, ρ, and a versus t.  Make sure you choose appropriate scales for the vertical axes of graphs.

Copy the graphs into your log.  What do they graphs you produced for the first second of the balloon's motion tell you?  How do y, v, ρ, and a evolve and why?  Answer in full sentences.

Now explore the motion of the balloon in a 2 h time interval after it has been released at t = 0.  Modify your program so that it stops after 2 h and writes the results of the calculations into a new row of the spreadsheet every 5 minutes.

Prepare sheet 2 of an Excel workbook as shown in the figure below.  All numbers in the spreadsheet represent physical quantities in SI units.
 A B C D E 1 t y v density a 2
On the menu bar click the developer tab and choose macro.  Choose a macro name, for example "balloon2", and click create.
A Visual Basic for Application (VBA) window will open.  It will contain two entries.
 Sub balloon2()End Sub
Enter the following program.
 Sub balloon2()Set w = Worksheets("sheet2")t = 0v = 0y = 0density = 1.225a = (density * 0.1372 - 0.098) / 0.01i = 0j = 2Do While t < 7200    If i = 300000 Then        a1\$ = "a" & j        b1\$ = "b" & j        c1\$ = "c" & j        d1\$ = "d" & j        e1\$ = "e" & j        w.Range(a1\$).Value = t / 60        w.Range(b1\$).Value = y        w.Range(c1\$).Value = v        w.Range(d1\$).Value = density        w.Range(e1\$).Value = a        i = 0        j = j + 1End Ift = t + 0.001y = y + v * 0.001 + 0.5 * a * 0.00001v = v + a * 0.001density = 1.225 * (1 - 0.00002257 * y) ^ 4.255a = (density * 0.1327 - 0.098 - 0.0177 * density * v * Abs(v)) / 0.01i = i + 1LoopEnd Sub Close the VBA window. Click on an empty cell in your worksheet.  On the menu bar Macro and run your program.   Watch how the program creates the spreadsheet.
Produce graphs of y, v, ρ, and a versus t.  The time t in the spreadsheet is now given in minutes.  Make sure you choose appropriate scales for the vertical axes of graphs.

Copy the graphs into your log.  What do they graphs you produced for 2 hours  of the balloon's motion tell you?  How do y, v, ρ, and a evolve and why?  What is happening after approximately 1.5 h?  Answer in full sentences.