Question

(a) What is the density of an object that is \(14 \%\) submerged when floating in water at \(0^{\circ} \mathrm{C} ?\) (b) What percentage of the object will be submerged if it is placed in ethanol at \(0^{\circ} \mathrm{C} ?\)

Short answer

(a) The object's density is 140 kg/m³. (b) 17.74% of the object will be submerged in ethanol.

Step-by-step solution

Understand Buoyant Force Concept

When an object floats in a fluid, the weight of the fluid displaced equals the weight of the object. This is known as the principle of buoyancy or Archimedes' principle. We will use this to find the density of the object.

Density of Water

The density of water at \(0^{\circ}C\) is known to be \( \rho_w = 1000 \, \text{kg/m}^3 \). We need this value to calculate the density of the object.

Calculate Object's Density for Part A

The percentage of the object submerged in water directly relates to its density relative to water. If the object is \(14\%\) submerged, the relative density is the same fraction of the object's density to water's density. Hence, \(\frac{\rho_o}{\rho_w} = 0.14\). We get \(\rho_o = 0.14 \times 1000 = 140 \, \text{kg/m}^3\).

Density of Ethanol

The density of ethanol at \(0^{\circ}C\) is approximately \( \rho_e = 789 \, \text{kg/m}^3 \). This will be used to calculate the submerged percentage in ethanol.

Calculate Percentage Submerged in Ethanol for Part B

Using a similar approach, set up the relation \(\frac{\rho_o}{\rho_e}\) to determine the percentage of the object submerged in ethanol. Substituting the known values, \(\frac{140}{789} \approx 0.1774\), which implies \(17.74\%\) of the object will be submerged in ethanol.

Key concepts

Buoyant Force

When an object is submerged in a fluid, it experiences an upward force known as buoyant force. This force acts in the opposite direction of gravity and is equal to the weight of the fluid displaced by the object.

Archimedes' Principle gives us a straightforward way to understand buoyancy:

• When an object floats, the buoyant force equals the weight of the object.
• If an object is partially submerged, only the volume of the submerged portion displaces water, creating a buoyant force.

For instance, if an object is 14% submerged in water, the buoyant force equals 14% of the weight of the water displaced by the entire volume of the object.

Density of Water

Density is an important property of water, especially when it comes to understanding buoyancy and floating objects. At 0°C, the density of water is particularly high at \[ \rho_w = 1000 \, \text{kg/m}^3 \].

This value is essential for calculating other densities when factors like temperature and state changes are controlled. In many physics problems, such as calculating how much of an object is submerged, we use this as a reference point. For most practical purposes, water is used as the standard for comparison because its density is reliable and consistent at a given temperature.

Density Calculations

Density calculations help determine whether an object will float or sink in a fluid. Density is defined as mass per unit volume: \[ \rho = \frac{m}{V} \], where \( m \) is mass, and \( V \) is volume.

In this specific exercise, knowing how much of the object is submerged helps us calculate its density relative to the fluid.

• For water at 0°C, if an object is 14% submerged, its density is 14% of the density of water.
• Thus, the object's density is calculated as \( \rho_o = 0.14 \times 1000 = 140 \, \text{kg/m}^3 \).

These calculations are essential to find how objects interact with fluids, predicting the extent of buoyancy or submersion.

Density of Ethanol

Ethanol is a commonly used fluid in experiments involving buoyancy because its density differs significantly from that of water. At 0°C, the density of ethanol is approximately \[ \rho_e = 789 \, \text{kg/m}^3 \].

This lower density compared to water means objects will submerge more in ethanol than in water, assuming the object's density is unchanged. For example, if an object's density calculated relative to water is known, we can determine how much of it will submerge in ethanol:

• If an object has a density of \( 140 \, \text{kg/m}^3 \), it submerges approximately \( \frac{140}{789} \approx 0.1774 \, \text{or} \, 17.74\% \).

This kind of calculation uses the same principles as calculations for any buoyant interaction, making ethanol a useful reference fluid for such exercises.