A sample of an unknown material appears to weigh 275 N in air and 160 N when immersed in alcohol of specific gravity 0.700. What is the volume of the material? What is the density of the material?
W(alc) =W(air)-F(buoyant) = W(air)-ρVg
V={W(air)-W(alc)}/ρg=
={275-160}/0.7•9.8 =16.8 m³
W(air) =mg =ρ₀Vg
ρ₀=W(air)/ Vg = 275/16.8•9.8 =1.67 kg/m³
To find the volume of the material and its density, we can use the following steps:
1. First, let's calculate the weight of the material when immersed in alcohol. The weight of the material in air is 275 N, and when immersed in alcohol, it becomes 160 N. The difference in weight is due to the buoyant force acting on the material in the fluid.
2. We know that the weight of an object in a fluid is equal to its weight in air minus the weight of the fluid displaced by the object. In this case, the weight of the fluid displaced is equal to the weight of the alcohol that has the same volume as the material. We can calculate the weight of the fluid displaced using the specific gravity of the alcohol.
- Specific gravity is the ratio of the density of a substance to the density of a reference substance. In this case, the reference substance is water, so the specific gravity of the alcohol is 0.700.
- Since the weight of the fluid displaced is equal to the weight of the alcohol, we can use the equation:
Weight of the fluid displaced = weight of the alcohol = density of the alcohol × volume of the material × acceleration due to gravity.
3. Rearranging the equation, we can solve for the volume of the material:
volume of the material = (weight of the fluid displaced) / (density of the alcohol × acceleration due to gravity).
4. Once we have the volume of the material, we can find its density using the formula:
density of the material = weight of the material in air / volume of the material.
Let's plug in the numbers to find the volume and density:
Weight of the fluid displaced = weight of the alcohol
= 275 N - 160 N
= 115 N
Density of the alcohol = specific gravity × density of water
= 0.700 × density of water
The density of water is approximately 1000 kg/m³, so:
Density of the alcohol = 0.700 × 1000 kg/m³
= 700 kg/m³
Acceleration due to gravity, denoted as "g", is approximately 9.8 m/s².
Now, we can calculate the volume of the material:
volume of the material = (weight of the fluid displaced) / (density of the alcohol × acceleration due to gravity)
= 115 N / (700 kg/m³ × 9.8 m/s²)
Finally, we can find the density of the material:
density of the material = weight of the material in air / volume of the material
= 275 N / volume of the material
Plug in these values to get the final answers.