. In a filled fluid container, the system’$ mass is 0.41 slugs. Its density is 30 Ibm/ft? & g-31.5 fps?. Find (a) specific volume (b) the total volume
The Correct Answer and Explanation is:
To solve this problem, we need to understand the relationship between mass, density, and volume. The formula for specific volume (vvv) is:v=1ρv = \frac{1}{\rho}v=ρ1
where:
- vvv is the specific volume,
- ρ\rhoρ is the density.
Given:
- Mass m=0.41 slugsm = 0.41 \, \text{slugs}m=0.41slugs,
- Density ρ=30 lbm/ft3\rho = 30 \, \text{lbm/ft}^3ρ=30lbm/ft3,
- Gravitational acceleration g=31.5 fps2g = 31.5 \, \text{fps}^2g=31.5fps2.
Part (a) Specific Volume
The specific volume is the volume occupied by a unit mass of the substance. We calculate it by using the given density.
Since the mass is in slugs and the density is given in lbm/ft³, we first need to convert the units of mass into lbm. We know that:1 slug=32.174 lbm.1 \, \text{slug} = 32.174 \, \text{lbm}.1slug=32.174lbm.
So:m=0.41 slugs×32.174 lbm/slug=13.2 lbm.m = 0.41 \, \text{slugs} \times 32.174 \, \text{lbm/slug} = 13.2 \, \text{lbm}.m=0.41slugs×32.174lbm/slug=13.2lbm.
Now, we can calculate the specific volume. We use the formula:v=VolumeMass=1ρ=130 lbm/ft3=0.0333 ft3/lbm.v = \frac{\text{Volume}}{\text{Mass}} = \frac{1}{\rho} = \frac{1}{30 \, \text{lbm/ft}^3} = 0.0333 \, \text{ft}^3/\text{lbm}.v=MassVolume=ρ1=30lbm/ft31=0.0333ft3/lbm.
Thus, the specific volume of the system is 0.0333 ft3/lbm0.0333 \, \text{ft}^3/\text{lbm}0.0333ft3/lbm.
Part (b) Total Volume
To find the total volume (VVV) of the system, we can use the formula:V=v×m.V = v \times m.V=v×m.
Substitute the values for specific volume and mass:V=0.0333 ft3/lbm×13.2 lbm=0.44 ft3.V = 0.0333 \, \text{ft}^3/\text{lbm} \times 13.2 \, \text{lbm} = 0.44 \, \text{ft}^3.V=0.0333ft3/lbm×13.2lbm=0.44ft3.
Thus, the total volume of the system is 0.44 ft30.44 \, \text{ft}^30.44ft3.
Explanation:
- The specific volume tells us how much space one unit mass (1 lbm) occupies. It is inversely related to density: the higher the density, the lower the specific volume.
- The total volume is simply the product of the specific volume and the total mass of the system.
