An undersea research chamber is spherical with an external diameter of 3.70 m. The mass of the chamber, when occupied, is 26000 kg. It is anchored to the sea bottom by a cable.
What is the tension in the cable?
To find the tension in the cable, we need to calculate the weight of the spherical chamber and then subtract the buoyant force acting on it.
1. Calculate the weight of the spherical chamber:
The weight is equal to the mass of the chamber multiplied by the acceleration due to gravity.
Weight = mass * gravity
Given:
Mass of the chamber (when occupied) = 26000 kg
Acceleration due to gravity = 9.8 m/s²
Weight = 26000 kg * 9.8 m/s² = 254,800 N
2. Calculate the buoyant force:
The buoyant force is equal to the weight of the water displaced by the chamber. For a completely submerged spherical chamber, the buoyant force can be calculated using Archimedes' principle.
The volume of a sphere is given by the formula:
Volume = (4/3) * π * r³
Since the chamber is completely submerged, it displaces an amount of water equal to its own volume.
Given:
External diameter of the chamber = 3.70 m
The radius of the chamber (r) is half of the diameter:
Radius = 3.70 m / 2 = 1.85 m
Volume = (4/3) * π * (1.85 m)³ ≈ 23.659 m³
The buoyant force can be calculated using the formula:
Buoyant force = density of water * volume * gravity
The density of water is approximately 1000 kg/m³.
Buoyant force = 1000 kg/m³ * 23.659 m³ * 9.8 m/s² ≈ 229,886 N
3. Calculate the tension in the cable:
The tension in the cable can be found by subtracting the buoyant force from the weight of the chamber.
Tension = Weight - Buoyant force
Tension = 254,800 N - 229,886 N ≈ 24,914 N
Therefore, the tension in the cable is approximately 24,914 N.