A balloon of negligible mass is filled with helium gas (p=0.179 kg/m^3) until its volume is just large enough to lift a 25.0 kg load. (a) Find this volume. (b) Find the buoyant force exerted by air (p=1.29 kg/m^3) on the balloon.
To find the volume of the balloon, we need to use the concept of buoyancy. Buoyancy is the upward force exerted on an object immersed in a fluid, in this case, air.
(a) To find the volume of the balloon:
Let's assume the volume of the balloon is V.
The density of helium gas is given as p = 0.179 kg/m^3, and the mass of the load is given as m = 25.0 kg.
When the balloon is just large enough to lift the load, its weight must be balanced by the buoyant force. Mathematically, this can be expressed as:
Weight of the load = Buoyant force
The weight of the load can be calculated using the equation:
Weight = mass × gravitational acceleration
Weight of the load = m × g
Where g is the acceleration due to gravity, which is approximately 9.8 m/s^2.
Now, let's consider the buoyant force. The buoyant force can be calculated using the equation:
Buoyant force = density of the fluid × volume × gravitational acceleration
Buoyant force = p_air × V × g
Where p_air is the density of air, given as 1.29 kg/m^3.
Since the weight of the load is balanced by the buoyant force, we can equate the two equations:
m × g = p_air × V × g
Simplifying and rearranging the equation, we can solve for the volume V:
V = (m × g) / (p_air)
Substituting the given values, we have:
V = (25.0 kg × 9.8 m/s^2) / (1.29 kg/m^3)
Now calculate the volume V to find the answer.
(b) To find the buoyant force exerted by air on the balloon:
We already have the volume of the balloon, so now we can calculate the buoyant force using the equation:
Buoyant force = p_air × V × g
Substitute the given values of p_air and the volume V calculated in part (a), and multiply by the gravitational acceleration g to find the buoyant force.