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

[anyone1979]

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

 

[kamerling]

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)

 

[Doc Al]

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 $R_o^3 - R_i^3$. 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.

 

[anyone1979]

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

 

[anyone1979]

OK.

This is the volume of the shell material (not the sphere). I assume you meant to write $R_o^3 - R_i^3$. 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

 

[anyone1979]

Sorry guys, I had a brain fart.
I just realized my mistake...You guys are both right.
Thanks for all your help

 

[Doc Al]

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:
$$V = \frac{4 \pi}{3} (R_{outer}^3 - R_{inner}^3)$$

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.

 

[anyone1979]

Thanks for all your help...