A 1.8-g balloon is filled with helium gas. When a mass of 4.8 g is attached to the balloon, the combined mass hangs in static equilibrium in midair. Assuming that the balloon is spherical, what is its diameter? (The density of helium is 0.179 kg/m3, and the density of air is 1.3 kg/m3.)
These problems can be solved using Archimedes principle. Show your work foor additional assistance.
To find the diameter of the balloon, we can use the concept of buoyancy. The buoyant force on the balloon is equal to the weight of the air it displaces, minus the weight of the balloon and the mass attached to it.
1. First, we need to find the volume of the balloon. We know the mass of the balloon, which is 1.8 grams. However, we need the mass in kg for our calculations. So we convert it to kg by dividing by 1000:
mass of balloon (in kg) = 1.8 g / 1000 = 0.0018 kg
2. We can use the density of helium to find the volume of the balloon using the formula:
volume of balloon = mass of balloon / density of helium
volume of balloon = 0.0018 kg / 0.179 kg/m^3
volume of balloon = 0.01006 m^3
3. Now, let's find the weight of the balloon and the mass attached to it. We know the mass attached to the balloon is 4.8 grams. Again, we convert it to kg:
mass attached to balloon (in kg) = 4.8 g / 1000 = 0.0048 kg
weight of balloon and attached mass = (mass of balloon + mass attached to balloon) * acceleration due to gravity
weight of balloon and attached mass = (0.0018 kg + 0.0048 kg) * 9.8 m/s^2
weight of balloon and attached mass = 0.0066 kg * 9.8 m/s^2
weight of balloon and attached mass = 0.06468 N
4. The buoyant force is equal to the weight of the air displaced by the balloon:
buoyant force = density of air * volume of balloon * acceleration due to gravity
buoyant force = 1.3 kg/m^3 * 0.01006 m^3 * 9.8 m/s^2
buoyant force = 0.128908 N
5. In static equilibrium, the buoyant force is equal to the weight of the balloon and attached mass. So we can set up the following equation:
weight of balloon and attached mass = buoyant force
0.06468 N = 0.128908 N
6. Now we can solve for the diameter of the balloon using the equation for the buoyant force on a sphere:
buoyant force = (4/3) * π * (radius of balloon)^3 * density of air * acceleration due to gravity
0.128908 N = (4/3) * π * (radius of balloon)^3 * 1.3 kg/m^3 * 9.8 m/s^2
(radius of balloon)^3 = 0.128908 N / [(4/3) * π * 1.3 kg/m^3 * 9.8 m/s^2]
(radius of balloon)^3 = 0.010317 m^3
7. Taking the cube root of both sides will give us the radius of the balloon:
radius of balloon = ∛(0.010317 m^3)
radius of balloon = 0.2154 m
8. Finally, we multiply the radius by 2 to get the diameter of the balloon:
diameter of balloon = 2 * radius of balloon
diameter of balloon = 2 * 0.2154 m
diameter of balloon = 0.4308 m