- The Mariana trench is located in the Pacific Ocean at a depth of about 11000 m below the surface of the water. The density of seawater is 1025 kg/m3. (a) If an underwater vehicle were to explore such a depth, what forces would the water exert on the vehicle’s observation window (radius = 10cm)? (b) For comparison, determinate the weight of a jetliner whose mass is 12x105kg.Take Patm = 1.013x105Pa
(a)To calculate the force exerted by the water on the observation window of the underwater vehicle, we can utilize the concept of pressure. The pressure at a certain depth in a fluid is given by:
P = ρgh
Where:
P = Pressure (in pascals)
ρ = Density of the fluid (in kg/m³)
g = Acceleration due to gravity (in m/s²)
h = Depth (in meters)
The force exerted on the window can be determined by multiplying the pressure by the area of the window.
F = P * A
Where:
F = Force (in newtons)
P = Pressure (in pascals)
A = Area (in square meters)
The area of the observation window can be calculated using the formula for the area of a circle:
A = πr²
Where:
A = Area (in square meters)
r = Radius (in meters)
Given:
Depth (h) = 11000 m
Density of seawater (ρ) = 1025 kg/m³
Radius of window (r) = 0.1 m (10cm)
First, let's calculate the pressure:
P = ρgh
P = 1025 * 9.8 * 11000
P ≈ 1.12 × 10^8 Pa
Next, let's calculate the area:
A = πr²
A = π * (0.1)^2
A ≈ 0.0314 m²
Finally, let's calculate the force:
F = P * A
F = 1.12 × 10^8 * 0.0314
F ≈ 3.52 × 10^6 N
So, the water would exert a force of approximately 3.52 × 10^6 newtons on the observation window of the underwater vehicle.
(b) To determine the weight of the jetliner, we can use the formula:
Weight = mass * acceleration due to gravity
Given:
Mass of the jetliner (m) = 12 × 10^5 kg
Acceleration due to gravity (g) = 9.8 m/s²
Pressure at sea level (Patm) = 1.013 × 10^5 Pa
Weight = m * g
Weight = 12 × 10^5 * 9.8
Weight ≈ 1.18 × 10^7 N
So, the weight of the jetliner is approximately 1.18 × 10^7 newtons.
To answer part (a) of your question, we can determine the forces exerted on the observation window of an underwater vehicle exploring the Mariana Trench.
The force exerted by a fluid (in this case, seawater) on an object immersed in it is governed by Archimedes' principle, which states that the buoyant force on an object is equal to the weight of the fluid it displaces.
In this case, the observation window is submerged in water, so the force exerted on it is the buoyant force. The buoyant force can be calculated using the equation:
Buoyant force = (density of water) * (volume of water displaced) * (acceleration due to gravity)
The density of seawater is given as 1025 kg/m3, and the volume of water displaced can be calculated as the volume of a cylinder:
Volume of water displaced = π * (radius of the window)2 * (height of the window)
The height of the window is the depth of the Mariana Trench, which is given as 11000 m.
Substituting the given values, we have:
Buoyant force = (1025 kg/m3) * π * (0.1m)2 * (11000 m) * (9.8 m/s2)
By evaluating this expression, you can find the force exerted by the water on the observation window.
To answer part (b) of your question, we need to determine the weight of a jetliner with a given mass. The weight of an object is the force exerted on it due to gravity and is given by the equation:
Weight = mass * acceleration due to gravity
The mass of the jetliner is given as 12x105 kg, and the acceleration due to gravity is approximately 9.8 m/s2.
Substituting the given values, we have:
Weight = (12x105 kg) * (9.8 m/s2)
By evaluating this expression, you can find the weight of the jetliner.
Note: In both calculations, it's important to ensure that the units are consistent and properly converted if necessary.