Suppose a distant world with surface gravity of 7.12 m/s2 has an atmospheric pressure of 7.48 104 Pa at the surface.
(a) What force is exerted by the atmosphere on a disk-shaped region 2.00 m in radius at the surface of a methane ocean?________N
(b) What is the weight of a 10.0-m deep cylindrical column of methane with radius 2.00 m? Note: The density of liquid methane is 415 kg/m3.
_____N
(c) Calculate the pressure at a depth of 10.0 m in the methane ocean.______Pa
(a) To find the force exerted by the atmosphere on a disk-shaped region, we can use the formula:
Force = Pressure x Area
The area of the disk-shaped region can be calculated using the formula:
Area = π x radius^2
Given:
Surface gravity = 7.12 m/s^2
Atmospheric pressure = 7.48 x 10^4 Pa
Radius of the disk-shaped region = 2.00 m
Let's calculate the force:
Area = π x (2.00 m)^2
Area = 4π m^2
Force = 7.48 x 10^4 Pa x 4π m^2
Force ≈ 939,754.2 N
Therefore, the force exerted by the atmosphere on the disk-shaped region at the surface of the methane ocean is approximately 939,754.2 N.
(b) To find the weight of a cylindrical column of methane, we can use the formula:
Weight = Density x Volume x Surface gravity
The volume of the cylinder can be calculated using the formula:
Volume = π x radius^2 x height
Given:
Density of liquid methane = 415 kg/m^3
Radius of the column = 2.00 m
Height of the column = 10.0 m
Let's calculate the weight:
Volume = π x (2.00 m)^2 x 10.0 m
Volume = 40π m^3
Weight = 415 kg/m^3 x 40π m^3 x 7.12 m/s^2
Weight ≈ 371,474.0 N
Therefore, the weight of the cylindrical column of methane with a depth of 10.0 m and a radius of 2.00 m is approximately 371,474.0 N.
(c) To calculate the pressure at a depth of 10.0 m in the methane ocean, we can use the formula:
Pressure = Pressure at the surface + (Density x Gravitational constant x Depth)
Given:
Surface pressure = 7.48 x 10^4 Pa
Density of liquid methane = 415 kg/m^3
Depth = 10.0 m
Let's calculate the pressure:
Pressure = 7.48 x 10^4 Pa + (415 kg/m^3 x 9.8 m/s^2 x 10.0 m)
Pressure ≈ 78,930 Pa
Therefore, the pressure at a depth of 10.0 m in the methane ocean is approximately 78,930 Pa.
To calculate the force exerted by the atmosphere on a disk-shaped region at the surface of a methane ocean, you can use the formula:
Force = Pressure * Area
(a) Firstly, let's calculate the area of the disk-shaped region with a radius of 2.00 m. The formula for the area of a circle is:
Area = π * r^2
where r is the radius.
Area = π * (2.00 m)^2 = 12.57 m^2
Now, we can calculate the force by multiplying the area by the atmospheric pressure:
Force = 7.48 * 10^4 Pa * 12.57 m^2 = 9.39 * 10^5 N
Therefore, the force exerted by the atmosphere on the disk-shaped region is approximately 9.39 × 10^5 N.
(b) To calculate the weight of a cylindrical column of methane, you can use the formula:
Weight = Density * Volume * Gravity
The volume of a cylinder is given by:
Volume = π * r^2 * h
where r is the radius and h is the height (or depth) of the cylinder.
Volume = π * (2.00 m)^2 * 10.0 m = 125.66 m^3
Now, we can calculate the weight by multiplying the volume by the density and gravity:
Weight = 415 kg/m^3 * 125.66 m^3 * 7.12 m/s^2 = 372,837.68 N
Therefore, the weight of the cylindrical column of methane is approximately 3.73 × 10^5 N.
(c) To calculate the pressure at a depth of 10.0 m in the methane ocean, you can use the formula:
Pressure = Atmospheric Pressure + (Density * Gravity * Depth)
Pressure = 7.48 * 10^4 Pa + (415 kg/m^3 * 7.12 m/s^2 * 10.0 m) = 1.06 * 10^5 Pa
Therefore, the pressure at a depth of 10.0 m in the methane ocean is approximately 1.06 × 10^5 Pa.