Question

Say the density of air is 1.23 kg/m3 and the density of helium is 0.164 kg/m3 when both are at STP. Let's model a balloon as a sphere of radius 0.12 m, with mass 2 g.

1) What is the mass of the helium that would fill a single balloon?

2) What is the buoyant force on a single balloon?

3) What is the net force on a single balloon?

4) If we attach 0.5 m of string (properties: 800 feet weighs 1 pound), what is the net force now on the balloon?

5) How many balloons would you need to have a 1 kg kitten in equilibrium, hovering just above the ground? (assume all of the strings are vertical, this is actually another problem)

Answer 1

given
density of air, rho_air = 1.23 kg/m^3
density of helium, rho_He = 0.164 kg/m^3
radius of the balloon, r = 0.12 m
mass of the balloon, m = 2 g = 0.002 kg

1) the mass of the helium that would fill a single balloon, m_He = rho_He*volume of balloon

= 0.164*(4/3)*pi*0.12^3

= 1.19*10^-3 kg <<<<<<<<---------------Answer

2) the buoyant force on a single balloon, B = weight of the displaced air by balloon

= rho_air*V_balloon*g

= 1.23*(4/3)*pi*0.12^3*9.8

= 8.72*10^-2 N <<<<<<<<---------------Answer

3) Net force on a single balloon = B - Weight of the balloon - weight of the He

= 8.72*10^-2 - 0.002*9.8 - 1.19*10^-3*9.8

= 0.0559 N <<<<<<<<---------------Answer

4) linear mass density of the string, mue = 1 pound/(800 feet)

= 1*0.45359 kg/(800*0.3048 m)

= 1.86*10^-3 kg/m

mass of the strung, m_string = mue*L

= 1.86*10^-3*0.5

= 0.00093 kg

Net force on a single balloon = B - Weight of the balloon - weight of the He - weight of the string

= 8.72*10^-2 - 0.002*9.8 - 1.19*10^-3*9.8 - 0.00093*9.8

= 0.0468 N <<<<<<<<---------------Answer

5) Number of balloon needed, N = weight of kitten/Net force on a single balloon

= 1*9.8/0.0468

= 209 <<<<<<<<---------------Answer