How do I find the mass of a sphere using proportionality?

Question

A steel sphere 3.0 cm in diameter has a mass of 0.11 kg. What is the mass of a steel sphere 5.0 cm in diameter?

I know that $m= DV$ where D is density and V is Volume. I am assuming that both spheres have the same density since it is made of steel so I plugged in $1.5$ into $V=\dfrac{4}{3}\pi r^3$ and got $14.14cm^3$. I then plugged $14.14$ into $m= DV$ and got that the density was $0.0078$. Now I can solve for the unknown mass $m=0.0078(\dfrac{4}{3}\pi (2.5)^3)$. I got the mass is $0.51kg$. Did I do this problem correctly? If not could anyone explain what I did wrong?

Answer 1

We are scaling linear dimensions (lengths) by the factor $\frac{5}{3}$.

If we scale linear dimensions by the factor $\lambda$, then volume is scaled by the factor $\lambda^3$.

So we are scaling volume, and therefore mass, by the factor $\left(\frac{5}{3}\right)^3$.

The new mass is therefore $\left(\frac{5}{3}\right)^3(0.11)$.

Remark: Your calculation was correct. More work, though! And suppose that the item was not a ball, but a small statue of a horse, and we were scaling linear dimensions by the factor $\frac{5}{3}$. The scaling argument above would still work, but there is no "volume of a horse" formula.

Answer 2

You said you wanted an answer using proportionality. I'll work out the proportion with algebra, only substituting at the end.

m = DV and V = 4πr³/3, so m = 4πDr³/3

Now take the ratio of two masses:

m₂/m₁ = (4πD₂r₂³/3)/(4πD₁r₁³/3)

Assuming they're the same kind of steel at the same ambient pressure and temperature, density is constant. This means D₂ = D₁. Now simplify the right side by dividing the top and bottom by 4πD₁/3:

m₂/m₁ = r₂³/r₁³

Now solve for m₂ by multiplying both sides by m₁:

m₂ = m₁ * r₂³/r₁³

Finally, substitute in m₁ = 0.11 kg, r₁ = 3.0 cm, and r₂ = 5.0 cm:

m₂ = 0.11 kg * (5.0 cm)³/(3.0 cm)³ = 0.51 kg