James Cameron piloted a submersible craft to the bottom of the Challenger Deep, the deepest point on the ocean's floor, 11,000 m below the surface. What was the total inward force on the 1.1-m-diameter pilot sphere in which Cameron sat?
To calculate the total inward force on the pilot sphere, we need to use the formula for pressure:
Pressure = Force / Area
Since the inward force is due to the pressure exerted by the water, we can rearrange the formula to solve for force:
Force = Pressure * Area
First, let's calculate the area of the pilot sphere:
Area = π * (radius)^2
The diameter of the sphere is given as 1.1 m, so the radius would be half of that:
Radius = 1.1 m / 2 = 0.55 m
Now we can calculate the area using the radius:
Area = π * (0.55 m)^2
Once we have the area, we can calculate the pressure at the bottom of the Challenger Deep. The pressure increases with depth, and we can use the formula:
Pressure = Density * Acceleration due to gravity * Depth
The density of seawater is about 1030 kg/m^3 and the acceleration due to gravity is approximately 9.8 m/s^2. The depth is given as 11,000 m. Let's substitute these values into the formula:
Pressure = 1030 kg/m^3 * 9.8 m/s^2 * 11000 m
Now we have the pressure, we can calculate the total inward force on the pilot sphere using the formula:
Force = Pressure * Area
Substituting in the values we calculated:
Force = (1030 kg/m^3 * 9.8 m/s^2 * 11000 m) * (π * (0.55 m)^2)
Calculating this expression will give us the total inward force on the 1.1-m-diameter pilot sphere.
To calculate the total inward force on the pilot sphere, we need to consider the pressure at the depth of the Challenger Deep. The pressure at any depth in a fluid can be calculated using the formula:
Pressure (P) = Density (ρ) * Gravity (g) * Depth (h)
In this case, the density of seawater is approximately 1030 kg/m^3, and the acceleration due to gravity is 9.8 m/s^2.
First, let's calculate the depth in meters. The Challenger Deep is 11,000 m below the surface.
Using the formula, we can calculate the pressure at the Challenger Deep:
Pressure = Density * Gravity * Depth
P = (1030 kg/m^3) * (9.8 m/s^2) * (11,000 m)
Now, we can calculate the total inward force on the pilot sphere assuming that the sphere is completely submerged and experiences uniform pressure on all sides. The inward force is equal to the pressure multiplied by the surface area of the sphere.
The surface area of a sphere is given by the formula:
Surface Area = 4 * π * Radius^2
The radius of the sphere is equal to half of the diameter (1.1 m).
Surface Area = 4 * π * (Radius)^2
Surface Area = 4 * π * (0.55 m)^2
Now, we can calculate the total inward force on the sphere:
Total Inward Force = Pressure * Surface Area
Substituting the values into the formula, we get:
Total Inward Force = (P) * (Surface Area)
Now, you can calculate the total inward force by plugging in the values into the equation.
the surface area of the sphere is ... π d^2
a cubic meter of water has a mass of 1000 kg (a metric ton)
... 11,000 m^3 of water has a mass of ... 1.1E7 kg
the force of the water (pressure) is ... 1.1E7 * g Newtons/m^2
... multiply by the surface area to find the total force