Calculate the density of the material of which the sphere is made

In summary: I think I got it now.In summary, the density of the material of the sphere is 1.6 * 10^3 kg/m^3. This was calculated by finding the weight of the sphere, which is equal to the weight of the displaced liquid, and using the volume of the shell material to calculate the density. The error in the previous calculation was due to incorrect volume calculation and not considering the buoyant force as the weight of the displaced liquid.
  • #1
anyone1979
38
0
A hollow sphere of inner radius 8.0 cm and outer radius 9.0 cm floats half submerged in a liquid of specific gravity 0.80. Calculate the density of the material of which the sphere is made.

Density of water = 1 * 10^3 kg/m^3
inner radius = 8.0 cm = 8 * 10 ^-2 m
outer radius = 9.0 cm = 9 * 10 ^-2 m
specific gravity = Density of liquid / density of water
Density of liquid = specific gravity * density of water = .80*1000 = 800 kg/m^3
V = (4 pi / 3) * (outer radius^3 * inner radius^3) = 9.09 *10 ^-4 m^3

weight of liquid = density of liquid * V * gravity = 7.127 N (buoyant force)

weight of sphere = weight of liquid + buoyant force = 7.127 + 7.127 = 14.254 N

mass of sphere = weight of sphere / gravity = 14.254/9.8 = 1.454 kg

density of sphere = mass of sphere / V = (1.454/(9.09 * 10^-4)) = 1.6 * 10^3 kg/m^3



-----All the work seems right, but I think the answer is not right. especially the calculation of the weight of the sphere. Can anybody help?





The Attempt at a Solution

 
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  • #2
I suppose the sphere is filled with air? In that case the volume of the displaced liquid is just (4 pi/3)*outer radius^3 but only half of the sphere is submerged so the displaced volume is (4 pi/3)*outer radius^3/2.

The weight of the displaced volume is equal to to total weight of the sphere.
the mass of the sphere is then easy.

to find the density you will then have to use the volume of only the shell (as you did)
 
  • #3
anyone1979 said:
A hollow sphere of inner radius 8.0 cm and outer radius 9.0 cm floats half submerged in a liquid of specific gravity 0.80. Calculate the density of the material of which the sphere is made.

Density of water = 1 * 10^3 kg/m^3
inner radius = 8.0 cm = 8 * 10 ^-2 m
outer radius = 9.0 cm = 9 * 10 ^-2 m
specific gravity = Density of liquid / density of water
Density of liquid = specific gravity * density of water = .80*1000 = 800 kg/m^3
OK.
V = (4 pi / 3) * (outer radius^3 * inner radius^3) = 9.09 *10 ^-4 m^3
This is the volume of the shell material (not the sphere). I assume you meant to write [itex]R_o^3 - R_i^3[/itex]. You'll need the volume of the sphere as well.

weight of liquid = density of liquid * V * gravity = 7.127 N (buoyant force)

weight of sphere = weight of liquid + buoyant force = 7.127 + 7.127 = 14.254 N
Not sure what you're doing here: The buoyant force equals the weight of the displaced fluid. What's the volume of fluid displaced?

And the buoyant force also equals the weight of the sphere, since its floating.

mass of sphere = weight of sphere / gravity = 14.254/9.8 = 1.454 kg

density of sphere = mass of sphere / V = (1.454/(9.09 * 10^-4)) = 1.6 * 10^3 kg/m^3
Once you properly calculate the weight of the sphere, this method will give you the density of the material.
 
  • #4
kamerling said:
I suppose the sphere is filled with air? In that case the volume of the displaced liquid is just (4 pi/3)*outer radius^3 but only half of the sphere is submerged so the displaced volume is (4 pi/3)*outer radius^3/2.

The weight of the displaced volume is equal to to total weight of the sphere.
the mass of the sphere is then easy.

to find the density you will then have to use the volume of only the shell (as you did)

Thanks, I just realized it...
I calculated all wrong
 
Last edited:
  • #5
Doc Al said:
OK.

This is the volume of the shell material (not the sphere). I assume you meant to write [itex]R_o^3 - R_i^3[/itex]. You'll need the volume of the sphere as well.


Not sure what you're doing here: The buoyant force equals the weight of the displaced fluid. What's the volume of fluid displaced?

And the buoyant force also equals the weight of the sphere, since its floating.


Once you properly calculate the weight of the sphere, this method will give you the density of the material.


Thanks...I realized my error now.
I made a mistake on the calculation
 
Last edited:
  • #6
Sorry guys, I had a brain fart.
I just realized my mistake...You guys are both right.
Thanks for all your help
 
  • #7
anyone1979 said:
This is the volume of the sphere material:
V = (4 pi / 3) * (outer radius^3 * inner radius^3) = 9.09 *10 ^-4 m^3
You mean:
[tex]V = \frac{4 \pi}{3} (R_{outer}^3 - R_{inner}^3)[/tex]

This is the volume of the sphere:
Vs = (4 pi / 3) * (outer radius^3) = 3.1 * 10 ^-3 m^3 = volume of displaced liquid
That's the volume of the sphere. Is the entire sphere submerged?

weight of liquid = density of liquid * V * gravity = (buoyant force)
Wl = 800 * (3.1 * 10 ^-3) *9.8 = 24.3 N
Correct this.

So what about the volume of the sphere material since I am trying to find the density of the material of the sphere...I was thinking I needed to use that volume.
You'll need the volume of shell material when calculating the density. First find the mass.
 
  • #8
Thanks for all your help...
 

Related to Calculate the density of the material of which the sphere is made

1. What is density and why is it important?

Density is a measure of how much matter is contained in a given volume of a substance. It is important because it can provide information about the composition and properties of a material.

2. How do you calculate the density of a material?

Density is calculated by dividing the mass of a material by its volume. The formula for density is D=m/v, where D is density, m is mass, and v is volume.

3. What units are used to measure density?

Density is typically measured in units of mass per volume, such as grams per cubic centimeter (g/cm3) or kilograms per cubic meter (kg/m3).

4. What factors can affect the density of a material?

The density of a material can be affected by factors such as temperature, pressure, and the arrangement of its molecules or atoms. It can also vary depending on the isotopes present in the material.

5. How does density relate to the buoyancy of an object?

Density plays a key role in determining an object's buoyancy, or its ability to float in a fluid. Objects with a higher density than the fluid they are placed in will sink, while objects with a lower density will float.

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