A helium balloon

Objective:

In this exercise 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.

Procedure:

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 r₀= 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 of the air is proportional to the pressure divided by the temperature (according to the ideal gas law). We therefore have r/r₀= PT₀/(P₀T) = [(T₀- ay)/T₀]^4.255= [(1 - 0.0002257y)]^4.255. Given r₀ 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 rVgj, where V = 0.014 m³ is the volume of the balloon. The buoyant force changes with altitude. The weight of the balloon is -mgj = -0.1176j. It is constant if we neglect small variations of g with altitude. The drag force has magnitude R = (1/2)DrAv²= 0.0177rv² if we use D = 0.5 and A = pr². Its direction is opposite to the direction of the velocity v. The total force acting on the balloon is F = rVgj - 0.1176j - 0.0177rvv. Its acceleration is a = F/m.

You will first explore the motion of the balloon in a 1s 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 1,225 kg/m³, and the acceleration of the balloon is F/m = (rVg - 0.1176)j/0.01 = (0.1372r - 0.099)/0.01j in SI units. In a short time interval Dt = 0.001 the altitude y changes Dy = (1/2)aDt², the velocity changes by Dv = aDt and the density changes to r = 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 tools, macro, macros. 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.

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 0.05s 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 tools, macro, macros, run. Watch how the program creates the spreadsheet.

Now produce graphs of y, v, r, and a versus t.

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.

On the menu bar click tools, macro, macros. 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("sheet1") t = 0 v = 0 y = 0 density = 1.225 a = (density * 0.1372 - 0.098) / 0.01 i = 0 j = 2 Do 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 + 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
	Close the VBA window.
	Click on an empty cell in your worksheet. On the menu bar click tools, macro, macros, run. Watch how the program creates the spreadsheet.

Now produce graphs of y, v, r, and a versus t. The time t in the spreadsheet is now given in minutes.

Questions:

	Interpret the graphs you produced for the first second of the balloons motion. How do y, v, r, and a evolve and why?
	Interpret the graphs you produced for 2 hours of the balloons motion. How do y, v, r, and a evolve and why?
	What is happening after approximately 1.5 h?

To earn extra credit add your name and e-mail address to your spreadsheet. In full sentences answer the questions posed above.

Save your Excel document (your name_exm2.xls) and attach it to an e-mail message to mbreinig@utk.edu.