The specific weight of water at ordinary pressure and temperature is 9.81 kN/m3. The specific gravity of mercury is 13.56. Compute the density of water and the specific weight and density of mercury.
The Correct Answer and Explanation is :
To solve this problem, we will calculate the density of water and the specific weight and density of mercury. We will use the following information:
- Specific weight of water: 9.81 kN/m³
- Specific gravity of mercury: 13.56
- Acceleration due to gravity (g): 9.81 m/s²
Step 1: Compute the density of water
The specific weight (( \gamma )) is related to the density (( \rho )) and gravitational acceleration (( g )) by the equation:
[
\gamma = \rho \times g
]
We are given the specific weight of water, so we can solve for the density of water:
[
\rho_{\text{water}} = \frac{\gamma_{\text{water}}}{g}
]
Substituting the known values:
[
\rho_{\text{water}} = \frac{9.81 \, \text{kN/m}^3}{9.81 \, \text{m/s}^2}
]
[
\rho_{\text{water}} = 1 \, \text{tonne/m}^3 = 1000 \, \text{kg/m}^3
]
Thus, the density of water is 1000 kg/m³.
Step 2: Compute the specific weight and density of mercury
The specific gravity of mercury is the ratio of its density to the density of water:
[
SG_{\text{mercury}} = \frac{\rho_{\text{mercury}}}{\rho_{\text{water}}}
]
We are given that the specific gravity of mercury is 13.56, so we can calculate the density of mercury:
[
\rho_{\text{mercury}} = SG_{\text{mercury}} \times \rho_{\text{water}}
]
Substituting the known values:
[
\rho_{\text{mercury}} = 13.56 \times 1000 \, \text{kg/m}^3 = 13560 \, \text{kg/m}^3
]
Now, we can calculate the specific weight of mercury using the formula:
[
\gamma_{\text{mercury}} = \rho_{\text{mercury}} \times g
]
Substituting the known values:
[
\gamma_{\text{mercury}} = 13560 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2
]
[
\gamma_{\text{mercury}} = 133,416.6 \, \text{N/m}^3 \approx 133.42 \, \text{kN/m}^3
]
Thus, the results are as follows:
- Density of water: 1000 kg/m³
- Specific weight of mercury: 133.42 kN/m³
- Density of mercury: 13560 kg/m³
Explanation:
- Density is a measure of how much mass is contained in a unit volume. Water has a density of 1000 kg/m³, meaning every cubic meter of water weighs 1000 kilograms.
- Specific weight is the weight per unit volume of a substance. It is related to density by the gravitational acceleration constant. Since mercury has a much higher specific gravity than water, its density and specific weight are significantly higher than that of water.