Density of a Compressed Sphere: Does Halving the Radius Double the Density?

[snubbly]

Air in a sphere has density "x" kg/m3

If radius is halved and air is compressed... does density double? ("2x" km/m3)

 

[kuruman]

One thing at a time. Suppose only one of these happened. Say the radius is halved. Does the density increase or decrease and by what factor? Once you figure that out, compress the air some more and see what happens to the final density relative to the initial.

 

[snubbly]

Radius halved - density would increase.

I don't know by what factor though...

I have the Q&A ... initial is 1.4 kg/m3 and once radius is halved, density is 11 kg/m3.

I have to show working and I just can't figure it out.

Multiplying 1.4 by 7.9 gives me the answer, but I don't understand.

 

[mg0stisha]

Multiplying 1.4 by 7.9 gives me the answer, but I don't understand.

What's the magical 7.9? Or did you just do 11 kg/m³ / 1.4kg/m³

 

[snubbly]

What's the magical 7.9? Or did you just do 11 kg/m³ / 1.4kg/m³

yes, just the 11 / 1.4.

clueless.

 

[mg0stisha]

What all are you given in the problem?

 

[snubbly]

This is the question from the textbook:

Air enclosed in a sphere has density p = 1.4 kg/m³. What will the density be if the radius of the sphere is halved, compressing the air within?

 

[mg0stisha]

What happens to the volume of a sphere if the radius is halved?

 

[darkmagic]

let say: D = Density
m = mass
v = volume

D₁ = 1.4 kg/m³

mass is constant since the air will only be compressed.

D = m/v yielding m = Dv

Therefore, D₁v₁=D₂v₂

but v of sphere is (3*pi*r³)/4

and r₂=r₁/2

You can get it now.

 

[mg0stisha]

Minor correction: volume of a sphere is $$V = \frac{4\pi r^{3}}{3}$$

 

[darkmagic]

Sorry. I interchanged 3 and 4.

 

[mg0stisha]

got your back ;)

 

[darkmagic]

Thanks.

 

[ideasrule]

A intuitive way to think about this: Think about a cube. If you cut the side lengths in half, what happens to volume? Halve one dimension and volume halves. Halve the second and volume decreases by a factor of 4. Halve the third and volume goes down how many times?

The same thing happens to a sphere.

 

[snubbly]

A intuitive way to think about this: Think about a cube. If you cut the side lengths in half, what happens to volume? Halve one dimension and volume halves. Halve the second and volume decreases by a factor of 4. Halve the third and volume goes down how many times?

The same thing happens to a sphere.

8.

Thank you, that really helped!