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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.

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.

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.

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.