Problem 19 - University Physics with Modern Physics

Question

An electrical short cuts off all power to a submersible diving vehicle when it is 30 \(\mathrm{m}\) below the surface of the ocean. The crew must push out a hatch of area 0.75 \(\mathrm{m}^{2}\) and weight 300 \(\mathrm{N}\) on the bottom to escape. If the pressure inside is 1.0 atm, what downward force must the crew exert on the hatch to open it?

Step-by-Step Solution

Answer

The crew must exert approximately 22,628 N downward force to open the hatch.

Step 1: Understand the Problem

We need to find the downward force required to open the hatch against water pressure, considering both the internal pressure and the weight of the hatch. The hatch is 30 meters below sea level, and the internal pressure is 1.0 atm.

Step 2: Calculate External Water Pressure

Use the formula for pressure due to a fluid: \( p = \rho g h \), where \( \rho \) is the density of seawater (approximately \( 1025 \text{ kg/m}^3 \)), \( g \) is the acceleration due to gravity (\( 9.81 \text{ m/s}^2 \)), and \( h \) is the depth (\( 30 \text{ m} \)). Calculate: \( p = 1025 \times 9.81 \times 30 \).

Step 3: Convert Atmospheric Pressure to Pascals

The internal pressure is 1.0 atm, which equals approximately 101,325 Pa. We will need this to calculate the net pressure.

Step 4: Calculate Net Pressure

The net pressure exerted by water is the difference between external water pressure and internal atmospheric pressure: \( \text{Net Pressure} = \text{External Pressure} - \text{Internal Pressure} \). Compute this using the external pressure calculated in Step 2 and the internal pressure from Step 3.

Step 5: Determine Force Required

The force required to counteract this net pressure and the weight of the hatch is given by \( F = (\text{Net Pressure} \times \text{Area}) + \text{Weight of the Hatch} \). With \( \text{Area} = 0.75 \text{ m}^2 \) and \( \text{Weight} = 300 \text{ N} \), compute the total force.

Key Concepts

Pressure Calculation

Understanding pressure is crucial, especially when dealing with situations like a submersible vehicle under the ocean, where pressure can impact survival. Pressure in a fluid is defined by the formula: \[ p = \rho g h \]where:

• \( \rho \): The density of the fluid (for seawater, it's approximately 1025 kg/m³)
• \( g \): The acceleration due to gravity (9.81 m/s²)
• \( h \): The depth below the fluid surface (in this problem, 30 m)

By calculating this, we find the water pressure exerted at the depth where the vehicle is submerged. It's necessary to keep in mind that this pressure increases with depth due to the weight of the water above.

Net Force

Net force is the sum of all forces acting on an object. When discussing a submersible vehicle, the net force involves calculating the difference between opposing pressures and incorporating any additional forces such as weight. In the given exercise, the net force required to open the hatch is calculated by:

• Finding the external pressure exerted by the water at 30 m depth
• Subtracting the internal atmospheric pressure (1 atm = 101325 Pa)

Once the net pressure is known, the force required to counter it can be found by considering the hatch's area and adding its weight (300 N). This ensures that the team knows precisely how much force they need to exert to escape.

Buoyancy

Buoyancy is a vital concept in fluid mechanics as it determines the upward force a fluid exerts on an object within it. Though not directly mentioned in the problem, this principle can support understanding submerged vehicles since buoyancy affects their movement in water.

It is influenced by the fluid's density and the submerged object's volume.

For buoyancy, think about how objects float more easily in denser fluids.

The force due to buoyancy doesn't directly contribute to the hatch's opening force but informs us about the ambient forces underwater. Grasping buoyancy helps students understand why divers, like the ones in the problem, experience different pressures and forces when submerged.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid in equilibrium at any point within the fluid due to the force of gravity. It's crucial for understanding situations involving submerged objects.

• This pressure increases with depth because the fluid's weight above exerts a force over the area.
• It's calculated independent of the fluid's movement (non-flowing products).

In the exercise, the hydrostatic pressure due to the ocean is integral to determining the force that must be overcome to open the hatch. This concept is fundamental to diving operations and underwater equipment, ensuring that engineers and scientists can accurately predict the conditions and necessary forces at various depths.