Problem 26 - Physics

Question

The vertical surface of a reservoir dam that is in contact with the water is \(120 \mathrm{~m}\) wide and \(12 \mathrm{~m}\) high. The air pressure is one atmosphere. Find the magnitude of the total force acting on this surface in a completely filled reservoir. (Hint: The pressure varies linearly with depth, so you must use an average pressure.)

Step-by-Step Solution

Answer

The total force on the dam is 84,672,000 N.

Step 1: Calculate Water Pressure at Any Depth

The pressure exerted by a fluid in an open container at a given depth is calculated using the formula: \( P = \rho g h \), where \( \rho \) is the density of the fluid (water, in this case), \( g \) is the acceleration due to gravity, and \( h \) is the depth below the water surface. For water, \( \rho \approx 1000 \, \text{kg/m}^3 \) and \( g \approx 9.8 \, \text{m/s}^2 \). However, we take an average depth for this problem.

Step 2: Determine Average Water Pressure

The average water pressure on the surface of the dam can be calculated using half the height of the water column. Thus, the average depth is \( h_{avg} = \frac{12}{2} = 6 \) m. Apply this to the pressure formula: \( P_{avg} = \rho g h_{avg} = 1000 \times 9.8 \times 6 = 58800 \, \text{Pa} \) (Pascal).

Step 3: Calculate Total Force on the Dam

Force is given by \( F = P \times A \), where \( A \) is the area of the surface in contact with the water. The area \( A \) of the dam is width \( \times \) height = \( 120 \times 12 \) m superscript{2}. Calculate the area: \( A = 1440 \, \text{m}^2 \). Thus, the total force is \( F = 58800 \times 1440 = 84672000 \, \text{N} \).

Step 4: Consider Atmospheric Pressure

Although atmospheric pressure acts on both sides of the dam, it effectively cancels itself out when calculating the difference in force acting specifically due to water pressure. Thus, we do not need to include atmospheric pressure in the calculation for this context.

Key Concepts

Force Calculation

The force exerted on a surface submerged in a fluid is a crucial aspect of understanding fluid dynamics. To calculate the total force acting on a vertical surface like a reservoir dam, we use the equation:

• Pressure (P): The force exerted per unit area by the fluid.
• Area (A): The total surface area in contact with the fluid.

The formula to find the force is: \[ F = P imes A \] In this scenario, the pressure (P) is determined by the water's depth, and the area (A) is the height multiplied by the width of the dam's vertical face in contact with water. In our case, the pressure is averaged over the dam's height. Once we have both measurements, multiplying them gives the total force the water exerts against the dam. This lets engineers determine how sturdy a dam must be to withstand the pressure from a fully filled reservoir.

Pressure Variation with Depth

The pressure exerted by a liquid varies with its depth due to the increasing weight of the water above. This is especially important in calculating pressures on submerged surfaces such as reservoir dams. As water depth increases, so does the pressure. It's given by the equation:\[ P = \rho g h \] where:

• \( \rho \) is the density of the liquid (for water, about 1000 kg/m³)
• \( g \) is the acceleration due to gravity (approximately 9.8 m/s²)
• \( h \) is the depth of the liquid

Typically, the pressure felt at the very bottom of a dam will be greater than the pressure closer to the surface.
However, since the problem specifies using an average pressure over the height, we use half of the dam's height. This approach simplifies calculations and still provides enough accuracy to determine total force. Understanding this variation is crucial for designing structures that are both efficient and safe.

Reservoir Dam

A reservoir dam is a critical infrastructure that holds back water while maintaining stability and safety. The water exerts hydrostatic pressure on the dam, which must be carefully calculated to ensure the structure can withstand the force. The vertical surface of the dam faces constant pressure variation due to changes in water depth. The primary considerations for engineers are:

• The height of the dam: Taller dams experience greater pressure at the bottom because of the weight of water above.
• The width of the dam: This affects the total area, which in turn affects the total force calculation.
• The material of the dam: Material strength is necessary to endure forces over long periods.

With the data presented, we can calculate the pressure exerted at an average depth and, using the formula for force, determine if the dam's materials can handle such stress. Design improvements might include increasing height or width, or using stronger materials to ensure long-term durability. Efficiently managing these factors keeps the community safe and ensures continuous water supply.