Problem 18 - Fundamentals of Physics

Question

An open tube of length \(L=1.8 \mathrm{~m}\) and cross-sectional area \(A=\) \(4.6 \mathrm{~cm}^{2}\) is fixed to the top of a cylindrical barrel of diameter \(D=1.2 \mathrm{~m}\) and height \(H=\) \(1.8 \mathrm{~m}\). The barrel and tube are filled with water (to the top of the tube). Calculate the ratio of the hydrostatic force on the bottom of the barrel to the gravitational force on the water contained in the barrel. Why is that ratio not equal to \(1.0 ?\) (You need not consider the atmospheric pressure.)

Step-by-Step Solution

Answer

The ratio of hydrostatic force to gravitational force is not 1.0 because hydrostatic pressure depends on height, not weight.

Step 1: Find Volume of Water

The volume of water in the barrel is the sum of the volume of the barrel and the tube. Calculate the volume of the barrel, which is a cylinder, using the formula \(V_{barrel} = \pi (\frac{D}{2})^2 H\), where \(D=1.2\, \mathrm{m}\) and \(H=1.8\, \mathrm{m}\). Substitute the values to get \(V_{barrel} = \pi (0.6)^2 \times 1.8\, \mathrm{m^3}\). Calculate the volume of the water in the tube using \(V_{tube} = A \times L\), where \(A = 4.6 \times 10^{-4}\, \mathrm{m^2}\), and \(L = 1.8\, \mathrm{m}\). Calculate both volumes and sum them to find the total volume of water.

Step 2: Calculate Gravitational Force

The gravitational force acting on the water is obtained using \(F_{gravity} = \rho g V\), where \(\rho=1000\, \mathrm{kg/m^3}\) is the density of water and \(g=9.8\, \mathrm{m/s^2}\). Use the total volume of water obtained in Step 1 to calculate \(F_{gravity}\).

Step 3: Calculate Hydrostatic Force

The hydrostatic force on the bottom of the barrel is the product of the pressure due to water at the bottom and the area of the barrel's base. The pressure \(P\) is \(P=\rho g h\), where \(h\) is the depth of water - the barrel and the tube combined (\(h = H + L\)). Consequently, the hydrostatic force \(F_{hydrostatic}\) is \(F_{hydrostatic} = P \pi (\frac{D}{2})^2\). Calculate this using the given dimensions and properties.

Step 4: Determine the Ratio

Find the ratio of hydrostatic force to gravitational force, \(\frac{F_{hydrostatic}}{F_{gravity}}\), using the values calculated in Steps 2 and 3. Show that this is different from 1.0.

Step 5: Explain the Ratio Discrepancy

The discrepancy in the ratio (\(\frac{F_{hydrostatic}}{F_{gravity}} eq 1\)) arises because the hydrostatic pressure depends only on the height of the water column, not its weight or volume. Meanwhile, gravitational force accounts for the entire weight of the water.

Key Concepts

Gravitational Force

When we discuss gravitational force, we're referring to the force exerted by gravity on an object due to its mass. In physics, we calculate it with an equation:

• Gravitational Force, \(F_{gravity}, \) is given by \( F_{gravity} = \rho g V \), where:
• \( \rho \) is the density of the fluid (for water, it's \(1000 \,\mathrm{kg/m^3}\)),
• \( g \) is the acceleration due to gravity (approximately \(9.8 \,\mathrm{m/s^2}\)), and
• \( V \) is the volume of the fluid.

This formula shows that gravitational force depends on the volume and density of the fluid and the gravitational pull. In the exercise, we used this formula to find the force acting on the water within the barrel and tube.
It's important to note that gravitational force represents the weight of the water, which directly influences how water behaves under gravity's effects.

Fluid Pressure

Fluid pressure is a fundamental concept crucial to solving problems involving liquids. It refers to the force exerted by a fluid per unit area and is influenced by the depth of the fluid. When calculating fluid pressure at a depth here's what we consider:

• Fluid pressure, \( P, \) can be calculated using \( P = \rho g h \), where:
• \( \rho \) is the density of the fluid,
• \( g \) is the gravitational acceleration, and
• \( h \) is the depth of the fluid column.

In the given problem, we considered the depth of water in both the barrel and the tube, which together influence the pressure at the base of the barrel. This depth leads to hydrostatic force being exerted downward. Keep in mind, fluid pressure doesn't depend on the volume or total mass of the water, only its depth.

Hydraulic Systems

Hydraulic systems take advantage of fluid pressure to transmit force from one point to another through a fluid-like water in our problem. Although the exercise did not directly deal with hydraulic systems, the principles are similar.

• In such systems, pressure is applied to one part of the fluid, transmitting it through the fluid to another location.
• For example, in the barrel problem, the water's weight creates pressure at its base, similar to force transmission in hydraulic systems.
• This pressure is then translated into a hydrostatic force, calculated at the barrel's bottom.

Understanding hydraulic systems helps you apply these principles to troubleshoot and design efficient systems in real-world applications, such as hydraulic brakes or lifting systems.

Physics Problem Solving

Successfully solving physics problems often uses a systematic approach to breaking down complex systems into simpler parts. Here are some tips:

• Understand what the problem is asking. In this exercise, the task was to compare hydrostatic vs. gravitational forces.
• Identify key equations and concepts, such as gravity's influence on fluid weight and fluid pressure at a given depth.
• Calculate step by step, as we did with volume, forces, and finally, their ratio.
• Interpret findings in context; understanding why the hydrostatic force differs from gravitational force due to depth's role instead of mass.

By following such a process, you're able to tackle complex physics problems with confidence, ensuring all aspects—like forces, pressure, and effects of fluid columns—are considered. Remember, practice enhances proficiency in physics problem-solving.