Problem 4 - Fundamentals of Physics

Question

Three liquids that will not mix are poured into a cylindrical container. The volumes and densities of the liquids are \(0.50 \mathrm{~L}, 2.6 \mathrm{~g} / \mathrm{cm}^{3} ;\) \(0.25 \mathrm{~L}, 1.0 \mathrm{~g} / \mathrm{cm}^{3}\); and \(0.40 \mathrm{~L}, 0.80 \mathrm{~g} / \mathrm{cm}^{3} .\) What is the force on the bottom of the container due to these liquids? One liter \(=1 \mathrm{~L}=\) \(1000 \mathrm{~cm}^{3}\). (Ignore the contribution due to the atmosphere.)

Step-by-Step Solution

Answer

The force on the bottom of the container is 18.326 N.

Step 1: Convert Volumes to cm³

The volumes of the liquids need to be converted from liters to cubic centimeters because the densities are given in grams per cubic centimeter. Use the conversion: \(1\, \text{L} = 1000\, \text{cm}^3\). - For the first liquid: \(0.50\, \text{L} = 0.50 \times 1000 = 500\, \text{cm}^3\)- For the second liquid: \(0.25\, \text{L} = 0.25 \times 1000 = 250\, \text{cm}^3\)- For the third liquid: \(0.40\, \text{L} = 0.40 \times 1000 = 400\, \text{cm}^3\)

Step 2: Calculate the Mass of Each Liquid

Use the formula \( \text{Mass} = \text{Density} \times \text{Volume} \) to find the mass of each liquid.- First liquid: \( \text{Mass} = 2.6\, \text{g/cm}^3 \times 500\, \text{cm}^3 = 1300\, \text{g} \)- Second liquid: \( \text{Mass} = 1.0\, \text{g/cm}^3 \times 250\, \text{cm}^3 = 250\, \text{g} \)- Third liquid: \( \text{Mass} = 0.80\, \text{g/cm}^3 \times 400\, \text{cm}^3 = 320\, \text{g} \)

Step 3: Calculate the Weight of Each Liquid

The weight of each liquid is calculated using the formula \( \text{Weight} = \text{Mass} \times \text{Acceleration due to gravity} \). The acceleration due to gravity \( g \) is approximately \( 9.8\, \text{m/s}^2 \).- First liquid: \( \text{Weight} = 1300\, \text{g} \times 9.8\, \text{m/s}^2 = 12,740\, \text{g} \cdot \text{m/s}^2 = 12.74\, \text{N} \) (since 1 N = 1 kg \(\cdot \) m/s² and 1 g = 0.001 kg)- Second liquid: \( \text{Weight} = 250\, \text{g} \times 9.8\, \text{m/s}^2 = 2,450\, \text{g} \cdot \text{m/s}^2 = 2.45\, \text{N} \)- Third liquid: \( \text{Weight} = 320\, \text{g} \times 9.8\, \text{m/s}^2 = 3,136\, \text{g} \cdot \text{m/s}^2 = 3.136\, \text{N} \)

Step 4: Calculate Total Force on the Bottom of the Container

The total force on the bottom of the container is the sum of the weights of all three liquids.- Total force: \(12.74\, \text{N} + 2.45\, \text{N} + 3.136\, \text{N} = 18.326\, \text{N} \)

Key Concepts

Density

Density is a measure of how much mass is contained in a given volume. It's a fundamental concept in fluid mechanics and is crucial in understanding how different substances behave.

The formula for density is simple: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). This means that the denser a substance is, the more mass it has in any given volume. For example, the first liquid in our exercise has a density of \(2.6\, \text{g/cm}^3\), which means it is more packed with mass compared to the third liquid with a density of \(0.80\, \text{g/cm}^3\).

Understanding density is important because:

• It helps predict how substances will layer when mixed.
• It influences the buoyancy of objects in a fluid.
• It affects the calculation of forces in fluid dynamics.

You might notice that denser liquids tend to sink below less dense ones. This layering can be clearly seen in various scientific and real-world applications. When solving problems related to fluid mechanics, first ensure that all density and volume measurements are in consistent units to avoid errors.

Force

Force is a push or a pull exerted on an object due to its interaction with another object. It is a vector quantity, which means it has both magnitude and direction. In the context of fluid mechanics, understanding the force exerted by a liquid, for example, is vital in various applications.

The most relevant force here is the weight of the liquids, which is a specific type of force caused by gravity pulling the mass of the liquid downwards. The formula to calculate this force is \( \text{Force} = \text{Mass} \times \text{Gravity} \).

Some points to remember about force:

• Measured in newtons (N) in the SI system.
• Can cause an object to start moving, stop, or change direction.
• In fluids, often related to pressure and surface area in contact.

In our exercise, calculating the force involves determining the weight of the liquids separately and then adding them to get the total force on the bottom of the container. This kind of force calculation is crucial for understanding how fluids interact with structures, which can be key in designing vessels, tanks, and ships.

Volume

Volume is the amount of space an object or substance occupies. In fluid mechanics, volume is crucial because it helps dictate how much fluid can fit into a given space, like a container. In our exercise, the volumes of the liquids are initially given in liters and then converted into cubic centimeters (\(\text{cm}^3\)), which is necessary for calculations.

The conversion between liters and cubic centimeters is straightforward:

• 1 liter = 1000 cubic centimeters (\(1000\, \text{cm}^3\)

Understanding volume is not only essential for conversion purposes but also because:

• It affects the buoyancy and pressure within a fluid.
• Determines the capacity or occupancy of a container.
• Informs decisions related to fluid flow rate and discharge.

In exercises like ours, proper volume conversion ensures that all subsequent calculations, like density and mass, are based on consistent units. This minimizes errors and allows you to correctly quantify how the properties of the liquid result in forces exerted on container surfaces.

Gravity

Gravity is the force by which a planet or other body draws objects toward its center. In fluid mechanics, gravity plays a crucial role in determining how fluids behave inside containers, particularly through its effect on weight.

The standard measure for gravity on Earth is approximately \(9.8\, \text{m/s}^2\), and it influences several aspects of fluid behavior:

• It causes all fluids to exert a weight force on the bottom of their containers.
• It affects free-falling objects and informs the sedimentation process.
• It initiates the pressure difference within a fluid column.

Gravity's influence in our exercise is seen in the way it pulls liquid masses downward, translating their mass into force via weight. Calculating the force due to gravity is essential to understand how much stress a liquid exerts on the bottom of its container, which is calculated by multiplying mass by the gravitational acceleration. This calculation is fundamental in fluid mechanics problems that involve assessing structural integrity and safety in vessels holding liquids.