Problem 43 - College Physics

Question

(a) If the density of an object is exactly equal to the density of a fluid, the object will (1) float, (2) sink, (3) stay at any height in the fluid, as long as it is totally immersed. (b) A cube \(8.5 \mathrm{~cm}\) on each side has a mass of \(0.65 \mathrm{~kg}\). Will the cube float or sink in water? Prove your answer.

Step-by-Step Solution

Answer

(a) Stay at any height; (b) Cube will sink.

Step 1: Understanding Buoyancy

For part (a), we need to apply the principle of buoyancy. When an object is placed in a fluid, it experiences an upward force called the buoyant force. If the object's density is equal to the fluid's density, then the buoyant force is equal to the weight of the object, allowing it to stay at any height as long as it's immersed.

Step 2: Calculation of Density

To solve part (b), we must calculate the density of the cube and compare it with the density of water. The density, \( \rho \), is given by the formula \( \rho = \frac{m}{V} \), where \( m \) is the mass and \( V \) is the volume. We have \( m = 0.65 \mathrm{~kg} \) and need to calculate \( V = (8.5^3) \mathrm{~cm^3} = 614.125 \mathrm{~cm^3} = 0.000614125 \mathrm{~m^3} \).

Step 3: Unit Conversion for Volume

Convert the volume from \( \mathrm{cm^3} \) to \( \mathrm{m^3} \) using the conversion factor \( 1 \mathrm{~cm^3} = 1 \times 10^{-6} \mathrm{~m^3} \). Thus, \( 614.125 \mathrm{~cm^3} = 0.000614125 \mathrm{~m^3} \).

Step 4: Density Calculation

Calculate the density using the formula \( \rho = \frac{m}{V} = \frac{0.65 \mathrm{~kg}}{0.000614125 \mathrm{~m^3}} \approx 1058.24 \mathrm{~kg/m^3} \).

Step 5: Comparison with Water Density

The density of water is about \( 1000 \mathrm{~kg/m^3} \). Since \( 1058.24 \mathrm{~kg/m^3} \) (density of cube) is greater than \( 1000 \mathrm{~kg/m^3} \) (density of water), the cube will sink.

Key Concepts

Density

The concept of density is fundamental in understanding why objects float or sink. Density is defined as the mass per unit volume of a substance and is mathematically expressed as \( \rho = \frac{m}{V} \), where \( \rho \) is density, \( m \) is mass, and \( V \) is volume.
For example, if you have two objects of the same size but different masses, the object with the higher mass will have a higher density. This can be visualized when comparing a piece of wood to a metal cube of the same size; typically, the metal is denser.
Understanding density helps in predicting whether an object will float or sink in a fluid, as it directly influences buoyant forces.

Density Calculation

Calculating the density of an object involves determining both its mass and its volume first. Let's take a cube with sides measuring 8.5 cm. To calculate its volume, you use the formula for the volume of a cube: \( V = s^3 \), where \( s \) is the length of a side. In this case, \( V = (8.5 \text{ cm})^3 = 614.125 \text{ cm}^3 \).
To convert this volume into cubic meters (a more commonly used metric unit), apply the conversion \( 1 \text{ cm}^3 = 10^{-6} \text{ m}^3 \). Thus, \( 614.125 \text{ cm}^3 = 0.000614125 \text{ m}^3 \).
With the mass given as 0.65 kg, we use the density formula \( \rho = \frac{m}{V} \) to find that \( \rho = \frac{0.65}{0.000614125} \approx 1058.24 \text{ kg/m}^3 \). This calculation indicates a relatively high density for the cube.

Principle of Buoyancy

Buoyancy refers to the upward force that a fluid exerts on an object placed in it. This force can make objects appear lighter and can even allow them to float. Archimedes' Principle states that the buoyant force on an object is equal to the weight of the fluid displaced by the object.
In practical terms, if a cube is immersed in water, it experiences this upward force. The interplay between the object's weight and the buoyant force determines its ability to float. Specifically:

• If the object's density is less than the fluid's, it will float.
• If the object's density is greater, it will sink.
• If the densities are equal, the object will remain suspended wherever placed in the fluid.

Understanding buoyancy is key to predicting whether the cube from the exercise will float or sink.

Comparison with Water Density

To determine whether the cube will float or sink, its density must be compared with the density of the fluid, typically water. The standard density of water is about \( 1000 \text{ kg/m}^3 \). When we calculated the cube's density as approximately \( 1058.24 \text{ kg/m}^3 \), we found it to be greater than that of water.
This comparison shows that the cube, being denser, will sink when placed in water. Objects only float when their density is less than that of the fluid, allowing the buoyant force to counteract their weight efficiently.
Thus, the exercise helps illustrate how a simple calculation and comparison can predict the behavior of objects in fluids.