A helium balloon lifts a basket and cargo of total weight 2250 N under standard conditions,
at which the density of air is 1.29 kg/m3 and
the density of helium is 0.178 kg/m3 .
What is the minimum volume of the
balloon? The acceleration of gravity is
9.81 m/s2 .
Answer in units of m3
To find the minimum volume of the balloon, we need to understand the principles of buoyancy. According to Archimedes' principle, an object submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the object.
In this case, the balloon is filled with helium, which is less dense than the surrounding air. This creates a buoyant force that opposes the weight of the balloon, basket, and cargo. For the balloon to float, the buoyant force must be greater than or equal to the weight of the system.
Let's calculate the buoyant force first. The buoyant force is given by the equation:
Buoyant force = density of fluid * volume of fluid displaced * acceleration due to gravity
Since the balloon is filled with helium, we'll use the density of helium for the density of the fluid. The volume of the fluid displaced is the same as the volume of the balloon (since the entire balloon is filled with helium).
Let's denote the minimum volume of the balloon as V. The buoyant force can be written as:
Buoyant force = (density of helium) * V * (acceleration due to gravity)
The weight of the system (balloon, basket, and cargo) is given as 2250 N. Therefore, for the balloon to float, the buoyant force must be equal to or greater than 2250 N.
Setting up the equation:
(density of helium) * V * (acceleration due to gravity) >= 2250 N
Now, we can substitute the values and solve for V:
0.178 kg/m^3 * V * 9.81 m/s^2 >= 2250 N
Simplifying the equation:
1.74438 V >= 2250
V >= 2250 / 1.74438
V >= 1289.03
Since the volume of a balloon cannot be negative, the minimum volume of the balloon required for it to lift the basket and cargo is approximately 1289.03 m^3.