Problem 77 - Master Resource Book in JEE Main Physics

Question

A tank is filled with water of density \(1 \mathrm{~g}\) per \(\mathrm{cm}^{\mathrm{3}}\) and oil of density \(0.9 \mathrm{~g} \mathrm{~cm}^{-3}\). The height of water layer is \(100 \mathrm{~cm}\) and of the oil layer is \(400 \mathrm{~cm}\). If \(g=980 \mathrm{cms}^{-2}\), then the velocity of efflux from an opening in the bottom of the tank is (a) \(\sqrt{900 \times 980} \mathrm{cms}^{-1}\) (b) \(\sqrt{1000 \times 980} \mathrm{cms}^{-1}\) (c) \(\sqrt{92 \times 980} \mathrm{cms}^{-1}\) (d) \(\sqrt{920 \times 980} \mathrm{cms}^{-1}\)

Step-by-Step Solution

Answer

The correct answer is (d) \(\sqrt{920 \times 980}\) cms\(^{-1}\).

Step 1: Understand the Problem

We need to find the velocity of efflux from an opening at the bottom of a tank that contains a layer of water topped by a layer of oil. The given densities are: water's density is \( 1 \text{ g/cm}^3 \) and oil's density is \( 0.9 \text{ g/cm}^3 \). The height of the water layer is 100 cm and the height of the oil layer is 400 cm. Gravity \( g \) is given as \( 980 \text{ cm/s}^2 \).

Step 2: Apply Torricelli’s Theorem

We will use Torricelli’s theorem which relates the velocity of efflux to the height of fluid in the tank. The equation is: \[ v = \sqrt{2gh} \]where \( h \) is the equivalent height of a single fluid calculated from the pressure due to water and oil.

Step 3: Calculate Equivalent Fluid Height

Calculate the effective height \( h' \) by considering the contribution of both water and oil. - Contribution from water layer: \( h_{water} = 100 \text{ cm} \)- Contribution from oil layer (adjusting density to water equivalence): \( h_{oil} = \frac{\text{Density of oil}}{\text{Density of water}} \times \text{Height of oil layer} = \frac{0.9}{1} \times 400 = 360 \text{ cm} \)The total effective height \( h' = h_{water} + h_{oil} = 100 \text{ cm} + 360 \text{ cm} = 460 \text{ cm} \)

Step 4: Calculate the Velocity of Efflux

Substitute the effective height into Torricelli's theorem:\[ v = \sqrt{2 \times 980 \times 460} \text{ cm/s} \]Simplifying gives:\[ v = \sqrt{920 \times 980} \text{ cm/s} \]

Step 5: Match with Given Options

The expression obtained for the velocity of efflux \( \sqrt{920 \times 980} \text{ cm/s} \) matches with option (d).

Key Concepts

Velocity of Efflux

In the world of fluid dynamics, the term **velocity of efflux** refers to the speed at which fluid exits a container through an opening. This concept is crucial for understanding and predicting fluid behavior in real-life situations, such as water flowing from a hole in a bucket.

To determine this speed, we often rely on Torricelli's Theorem, which connects the velocity of efflux to the height of the fluid in the container. The theorem yields the formula: \[ v = \sqrt{2gh} \] where

• \( v \) is the velocity of efflux,
• \( g \) represents the acceleration due to gravity,
• \( h \) is the height of the fluid above the opening.

This equation arises from principles of energy conservation, implying that the potential energy of the fluid is converted into kinetic energy as it exits the container.

In scenarios involving mixed fluids, such as a tank containing layers of water and oil, calculating the effective height becomes crucial, as it contributes to the hydrostatic pressure impacting the efflux velocity.

Hydrostatic Pressure

Hydrostatic pressure plays a vital role in determining the velocity of efflux, as it accounts for the pressure exerted by a fluid due to its weight. In simple terms, it is the force exerted by a static fluid on a surface when the fluid is at rest.

For fluids with multiple layers, like our tank with water and oil, the total hydrostatic pressure results from the sum of pressures due to each distinct layer. Each layer's pressure contribution depends on its height and density.

For example, the contribution to hydrostatic pressure from the oil in the tank is calculated by adjusting its height relative to an equivalent fluid (water in this case) using density ratios. Purely, the hydrostatic pressure from each layer is given by: \[ P = \rho gh \] where

• \( \rho \) is the fluid density,
• \( g \) is the gravitational acceleration,
• \( h \) signifies the height of the fluid layer.

Understanding this helps us correctly determine the effective combined pressure exerted by different fluids, which is then used in conjunction with Torricelli's theorem to find the actual efflux velocity.

Fluid Density Effects

Fluid density is a fundamental property influencing both hydrostatic pressure and fluid. The **density** of a fluid refers to its mass per unit volume, which directly affects how much pressure it can exert at a given depth.

When dealing with different fluids, such as water and oil, each possessing distinct densities, it becomes pivotal to account for these variations to calculate the efflux velocity accurately. A fluid with higher density will exert more pressure, given the same height, compared to a less dense fluid.

In the given tank problem, water's density (\( 1 ext{ g/cm}^3 \)) and oil's density (\( 0.9 ext{ g/cm}^3 \)) must be factored into the measurement of the effective fluid height. This involves converting the oil layer using the density ratio to reflect its equivalent pressure in water terms, making calculations in terms of a single reference density possible.

Thus, effectively accounting for these **fluid density effects** ensures a correct representation of the pressure at the tank's bottom, which is critical for calculating the velocity accurately via Torricelli’s theorem.