2.- A man of 73 kg has a volume of 75 l, of which 5 l correspond to the lungs. (a) Calculate the volume of the lungs when going in apnea underwater to a depth of 10 m, assuming that the density of sea water is 103 kg/m3 . (b) At that depth, will the total forces acting on him make him sink or will he be pushed towards the surface?
(a) To calculate the volume of the lungs when going in apnea underwater to a depth of 10 m, we need to consider the change in pressure at that depth and use the ideal gas law.
We start by calculating the change in pressure from the surface to a depth of 10 m using the formula:
ΔP = ρgh
Where:
ΔP is the change in pressure
ρ is the density of sea water (given as 103 kg/m3)
g is the acceleration due to gravity (approximately 9.8 m/s2)
h is the depth (10 m)
Substituting the given values into the formula, we have:
ΔP = (103 kg/m3) * (9.8 m/s2) * (10 m)
ΔP = 10,090 Pa
Next, we use the ideal gas law equation to relate pressure, volume, and temperature:
P1 * V1 / T1 = P2 * V2 / T2
Assuming that the temperature remains constant, we can rearrange the equation to solve for V2, the final volume of the lungs:
V2 = (P1 * V1 * T2) / (P2 * T1)
Since the pressure and volume of the lungs at the surface are given as 1 atm and 5 liters respectively, we can plug in the values:
V2 = (1 atm * 5 L * 1) / ((1 atm + 10,090 Pa) * 1)
Converting 1 atm to Pascals (Pa), we have:
1 atm = 101,325 Pa
V2 = (101,325 Pa * 5 L) / ((101,325 Pa + 10,090 Pa) * 1)
V2 = 505,650 Pa*L / 111,415 Pa
V2 ≈ 4.54 L
Therefore, the volume of the lungs when going in apnea underwater to a depth of 10 m is approximately 4.54 liters.
(b) To determine if the man will sink or be pushed towards the surface at that depth, we need to analyze the forces acting on him.
There are two major forces acting on the man underwater: the gravitational force and the buoyant force.
The gravitational force (weight) is given by the formula:
Weight = mass * g
In this case, the man's weight is calculated based on his mass of 73 kg:
Weight = 73 kg * 9.8 m/s2
Weight ≈ 715.4 N
The buoyant force is given by the formula:
Buoyant force = density of fluid * volume displaced * g
In this case, the density of sea water is given as 103 kg/m3 and the volume displaced is the total volume of the man minus the volume of his lungs submerged:
Volume displaced = (total volume - lung volume when submerged)
Substituting the values into the equation, we have:
Buoyant force = 103 kg/m3 * (75 L - 4.54 L) * 9.8 m/s2
Buoyant force ≈ 721.9 N
Comparing the weight and the buoyant force, we find that the buoyant force exceeds the weight:
Buoyant force > Weight
Therefore, the total forces acting on the man will make him float or be pushed towards the surface at that depth.