Answer :
To determine the relative density (also known as specific gravity) of a material, we can use the concept of buoyancy and Archimedes' principle. The relative density is a measure of how much denser a substance is compared to water.
Given:
- The weight of the material in air: 98.1 kg-m/s² (which is equivalent to its weight in newtons since 1 kg-m/s² is equal to 1 N).
- The weight of the material when submerged in water: 58.86 kg-m/s².
The loss in weight when the object is submerged in water is due to the buoyant force, which is equal to the weight of the water displaced by the object. This can be used to find the volume of the object and its density compared to water.
Step-by-step Solution:
Calculate the Buoyant Force:
The buoyant force experienced by the object when submerged in water can be calculated as the difference in its weight in air and its apparent weight in water.
[tex]F_{\text{buoyant}} = W_{\text{air}} - W_{\text{water}} = 98.1 \text{ N} - 58.86 \text{ N} = 39.24 \text{ N}[/tex]
Find the Volume of Displaced Water:
Since the buoyant force is equal to the weight of the displaced water, and knowing the density of water to be approximately 1000 kg/m³, we use the formula:
[tex]F_{\text{buoyant}} = \text{Volume}_{\text{water displaced}} \times 1000 \text{ kg/m}^3 \times g[/tex]
Given that the gravitational force, [tex]g = 9.81 \text{ m/s}^2[/tex]:
[tex]39.24 = \text{Volume}_{\text{water displaced}} \times 1000 \times 9.81[/tex]
Solving for the volume:
[tex]\text{Volume}_{\text{water displaced}} = \frac{39.24}{1000 \times 9.81} \approx 0.004 \text{ m}^3[/tex]
Calculate the Density of the Material:
The density of the material can be calculated as its mass divided by its volume. The mass can be found from its weight in air divided by gravity:
Mass of the object:
[tex]m = \frac{W_{\text{air}}}{g} = \frac{98.1}{9.81} \approx 10 \text{ kg}[/tex]
Density of the material:
[tex]\rho_{\text{material}} = \frac{10}{0.004} = 2500 \text{ kg/m}^3[/tex]
Calculate Relative Density:
The relative density is the ratio of the density of the material to the density of water.
[tex]\text{Relative Density} = \frac{\rho_{\text{material}}}{\rho_{\text{water}}} = \frac{2500}{1000} = 2.5[/tex]
Therefore, the relative density of the material is 2.5. This means the material is 2.5 times denser than water.
