Calculate the volume needed to have a weight of 390 lbf.
The correct answer and explanation is :
To calculate the volume needed to have a weight of 390 pounds-force (lbf), we need to consider the density of the material we are working with, since weight is a force that depends on both the mass and the gravitational acceleration acting on it.
The formula for weight is:
[
\text{Weight (W)} = \text{Mass (m)} \times \text{Gravitational acceleration (g)}
]
Where:
- Weight (W) is measured in pounds-force (lbf)
- Mass (m) is measured in pounds-mass (lbm)
- Gravitational acceleration (g) on Earth is approximately ( 32.174 \, \text{ft/s}^2 )
But we are interested in the volume required to achieve a specific weight, which depends on the material’s density. The mass is related to the volume (V) by:
[
\text{Mass (m)} = \text{Density (ρ)} \times \text{Volume (V)}
]
Thus, to calculate the volume for a given weight, we combine the equations. We first solve for mass from the weight equation:
[
\text{Mass (m)} = \frac{\text{Weight (W)}}{g}
]
Then substitute this expression for mass into the density-volume equation:
[
\text{Volume (V)} = \frac{\text{Mass (m)}}{\text{Density (ρ)}} = \frac{\frac{\text{Weight (W)}}{g}}{\text{Density (ρ)}}
]
Now, we will calculate the volume for 390 lbf, but we still need to know the density of the material. Let’s assume the material has a density of 62.4 lbm/ft³, which is the density of water (since the density is typically given in pounds-mass per cubic foot for the imperial system). The gravitational acceleration, ( g ), is 32.174 ft/s².
Now, we can plug in the values:
[
\text{Volume (V)} = \frac{390}{32.174 \times 62.4}
]
Let’s calculate the volume.
The volume needed to achieve a weight of 390 lbf, assuming the material has a density of 62.4 lbm/ft³ (like water), is approximately 0.194 cubic feet.
Explanation:
- Weight (390 lbf): The total weight you want to achieve.
- Gravitational acceleration (32.174 ft/s²): This is the acceleration due to gravity at Earth’s surface.
- Density (62.4 lbm/ft³): This is the density of the material, in this case, assumed to be water. If you were using a different material, the density would change, and so would the volume required.
To find the volume that corresponds to 390 lbf, we first calculate the mass in pounds-mass (lbm) using the formula:
[
\text{Mass (m)} = \frac{\text{Weight (W)}}{g}
]
This gives us the mass in pounds-mass, which we then convert into volume by dividing by the material’s density. For the given assumptions, the result is about 0.194 cubic feet.