Problem 49 - Master Resource Book in JEE Main Physics

Question

Water of density \(\rho\) at a depth \(h\) behind the vertical face of dam whose cross-sectional length is \(\lambda\) and cross-sectional area \(A\). It exerts a horizontal resultant force on the dam tending to slide it along its foundation and a torque tending to overturn the dam about the point \(O_{\pm}\) Pressure on the vertical face of the dam is (a) \(\rho g h\) (b) \(\frac{1}{2} \rho g h\) (c) \(\rho g h^{2}\) (d) \(\frac{1}{2} \rho g h^{2}\)

Step-by-Step Solution

Answer

The correct choice for the pressure is (b) \( \frac{1}{2} \rho g h \).

Step 1: Understanding Fluid Pressure

Pressure exerted by a fluid at a certain depth is given by the formula \( P = \rho g h \), where \( \rho \) is the fluid density, \( g \) is the acceleration due to gravity, and \( h \) is the depth of the fluid.

Step 2: Applying to Problem

In the context of this problem, water pressure on the vertical face of the dam varies linearly with depth. At the surface, the pressure is zero, and at depth \( h \), the pressure becomes \( \rho g h \).

Step 3: Calculating Average Pressure

Since the pressure increases linearly from zero at the top to \( \rho g h \) at the bottom, the average pressure across the height of the dam is half of \( \rho g h \), which is \( \frac{1}{2} \rho g h \).

Step 4: Formulating the Result

Since the problem specifically asks for the pressure on the vertical face, and we've calculated the average pressure as \( \frac{1}{2} \rho g h \), the correct choice for the pressure exerted is (b) \( \frac{1}{2} \rho g h \).

Key Concepts

Pressure in Fluids

The concept of pressure in fluids is foundational in understanding how fluids exert force on surfaces. Pressure is defined as the force applied per unit area. In fluids, this pressure depends on several factors. It is crucial to understand these factors to analyze how fluid pressure behaves on submerged surfaces.

Fluids rest or flow due to their density, which is the mass per unit volume. Density, denoted by \( \rho \), is a key player in determining pressure in fluids. Additionally, gravity \( g \), which accelerates objects downwards, also heavily influences fluid pressure.

Fluid pressure increases with depth due to the weight of the additional fluid above. This pressure is given by the formula \( P = \rho g h \), where \( h \) is the depth from the surface. This formula shows that, at any given depth, the pressure in a fluid is the same in all directions, a principle known as hydrostatic pressure.

• Thus, the deeper you dive into the fluid, the more pressure you experience from the fluid's weight above you.
• Understandably, at the fluid's surface, the pressure is minimal and increases steadily with depth.
• This behavior of pressure is critical when analyzing the stability and strength of structures like dams which hold back large bodies of water.

Forces on Dams

Dams face a considerable amount of force because they hold back large volumes of water. The critical forces at play mainly arise from the fluid pressure exerted by the water at different depths. This pressure results in a horizontal force acting on the dam's walls.

The total force exerted by water on a dam is a function of the water's density, gravity, and depth. Given the fluid pressure equation \( P = \rho g h \), and knowing that pressure increases linearly with depth, one can deduce that force is maximized at the base of the dam, where depth is greatest.

To calculate the force correctly, we consider the average pressure over the height of the dam, which is half of the maximum pressure at the base of the dam. Thus, the average pressure is \( \frac{1}{2} \rho g h \). This average pressure leads to a net force that acts horizontally against the dam structure.

• This force can cause sliding along the foundation and needs careful engineering to counter.
• Additionally, a torque effect may also be present, which attempts to rotate and overturn the dam.
• Thus, structural reinforcement and load distribution are essential in dam design to prevent structural failures.

Fluid Mechanics in Physics

Fluid mechanics is the branch of physics concerned with the behavior of fluids (liquids, gases, and plasmas) and the forces acting on them. It plays a vital role in understanding phenomena like fluid pressure and the forces exerted by fluids on structures.

In fluid mechanics, principles such as the Bernoulli's equation or Pascal's law help in predicting how fluids will behave under various conditions. For instance, understanding fluid flow helps in designing systems for water transport or irrigation.

Engineering structures like dams rely on the principles of fluid mechanics. Fluid mechanics considerations ensure that these structures withstand the immense forces exerted by fluid pressures.

• Engineers calculate anticipated forces using the pressure-like \( P = \rho g h \) and make informed choices in material and design to ensure safety.
• Due to the linear nature of pressure with respect to depth, solutions often integrate knowledge from fluid dynamics and statics.
• These principles ensure that structures withstand not just constant pressure but also variable or sudden changes such as floods or high tides.

Fluid mechanics remains one of the core applications of physics, ensuring sound engineering practices in both everyday and monumental structures.