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For Problems \(8-9\) A cylindrical object of cork of mass \(15 \mathrm{~g}\) and cross-sectional area \(A_{1}=10\) \(\mathrm{cm}^{2}\) floats in a pan of water as shown in the figure. An aluminium cylinder of mass \(25 \mathrm{~g}\) and cross-sectional area \(A_{2}=2 \mathrm{~cm}^{2}\) is attached \(4 \mathrm{~cm}\) below the cork and slides through a watertight frictionless hole in the bottom of the pan. Take density of the cork, \(\rho=0.2\) \(\mathrm{g} / \mathrm{cm}^{3}, \rho_{\text {aluminium }}=2.7 \mathrm{~g} / \mathrm{cm}^{3}, g=10 \mathrm{~m} /\) \(\mathrm{s}^{2}\). Based on the above information, answer the following questions. The length of the cork cylinder inside the water in equilibrium is (1) \(6 \mathrm{~cm}\) (2) \(4 \mathrm{~cm}\) (3) \(8 \mathrm{~cm}\) (4) \(3 \mathrm{~cm}\)

Step by step solution

01

Determine the Volume of Cork

First, we calculate the volume of the cork using its mass and density. The formula for density is \( \rho = \frac{m}{V} \), where \( m \) is mass and \( V \) is volume. For the cork, the volume \( V_1 = \frac{15 \text{ g}}{0.2 \text{ g/cm}^3} = 75 \text{ cm}^3 \).

02

Calculate the Submerged Volume

Since the cork is floating, it must displace an amount of water equal to its weight. The buoyant force \( F_b = \text{weight of the cork} = mg \). Substituting the value, \( F_b = 15 \text{ g} \times 10 \text{ m/s}^2 = 150 \text{ g cm/s}^2 \approx 1.5 \text{ N} \) (consider \( g \) in \( cm/s^2 \) which equals to 1000 \text{ cm/s}^2). Since the area of the cork \( A_1 = 10 \text{ cm}^2 \), \[ V_s = \frac{1.5 \text{ N}}{1 \text{ cm}^2 \times 1 \text{ cm}^2} = 1.5 \text{ cm}^3 \text{ (volume of water displaced by submerged part)} \].

03

Compute Length of Cork Submerged

Using the submerged volume of the cork \( V_s = 1.5 \text{ cm}^3 \) and its cross-sectional area \( A_1 = 10 \text{ cm}^2 \), the submerged length \( h \) can be determined: \( V_s = A_1 \times h \Rightarrow h = \frac{1.5 \text{ cm}^3}{10 \text{ cm}^2} = 0.15 \text{ cm} \).

04

Consider Aluminum's Effect

The aluminum attached 4 cm below does not affect the buoyant calculation directly as it passes through a frictionless, watertight hole. However, aluminum's presence means additional weight which indirectly affects equilibrium by needing to displace a total weight of the entire setup (cork + aluminum mass): Thus calculation becomes as balancing forces (not just buoyancy but total weight). By adjusting values through solving adjustments between additional force aluminum mass exerts and what cork can displace effectively maintaining 'effective equilibrium.' Solution simplifies back to identifying submerged length feasible sans hole impact directly.

05

Final Solution Calculation

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Buoyancy

Buoyancy refers to the ability of an object to float in a fluid, such as water. When an object is submerged in a fluid, it displaces some of the fluid, causing it to exert an upward force on the object. This upward force is known as the buoyant force. The magnitude of this force is equal to the weight of the displaced fluid, which is determined by the volume of fluid displaced and its density.
A floating object is in equilibrium when the buoyant force equals its weight. If the object weighs more than the displaced fluid's weight, it will sink; if less, it will float. In our exercise, the cork floats because the buoyant force balances its weight.

Density

Density plays a crucial role in determining whether an object will float or sink in a fluid. It is defined as a mass per unit volume, given by the formula: \[ \rho = \frac{m}{V} \]where \( \rho \) is the density, \( m \) is the mass, and \( V \) is the volume.
Different materials have different densities. For example, the density of cork is \(0.2 \text{ g/cm}^3 \) compared to the water it floats on. This low density allows cork to easily displace enough water to float. In contrast, materials like aluminum, with a higher density of \(2.7 \text{ g/cm}^3 \), will sink unless assisted by additional buoyant forces.

Equilibrium

Equilibrium in fluid mechanics is achieved when the net force acting on an object is zero. For floating objects, this means the upward buoyant force must balance the downward force of gravity. When these forces are equal, the object neither sinks nor rises; it remains floating at a constant position.
In this problem, the cork's equilibrium is when the buoyancy force, which lifts it, equals the total weight of the cork and effectively accounts for any interaction with attached components, like the aluminum. Thus, for equilibrium, the displaced water's weight must equal this total weight.

Archimedes' Principle

Archimedes' Principle is a key concept in understanding buoyancy. It states that a body wholly or partially submerged in a fluid experiences an upward force equal to the weight of the fluid displaced by it. This principle explains why ships float and submarines submerge.
In our exercise, Archimedes' Principle helps explain why the cork floats. As it enters the water, it displaces a volume of water equal to its own weight. Because the cork's weight is less than the weight of the displaced water, it achieves buoyancy and thus floats. Archimedes' Principle not only explains how objects float but also helps calculate the buoyant force essential for solving problems related to floating bodies.