Let ρAl represent the density of aluminum and ρFe that of iron. Find the radius of a solid aluminum sphere that balances a solid iron sphere of radius rFe on an equal-arm balance. (Use any variable or symbol stated above as necessary.)

Because the balancing arm is equal, therefore the masses of the aluminum and iron spheres are equal.

Define
[tex]r_{A}[/tex] = radius of aluminum sphere
[tex]r_{I}[/tex] = radius of iron sphere
[tex]\rho _{A}[/tex] = density of aluminum
[tex]\rho _{I}[/tex] = density of iron

Therefore
[tex] \frac{4 \pi }{3} r_{A}^{3} \rho _{A} = \frac{4 \pi }{3} r_{I}^{3} \rho _{I} \\\\ r_{A}^{3} = \frac{\rho _{I}}{\rho_{A}} r_{I}^{3} \\\\ r_{A} = r_{I} ( \frac{\rho _{I}}{\rho _{A}})^{1/3} [/tex]

Answer: [tex]r_{A} = r_{I} ( \frac{\rho _{I}}{\rho _{A}} )^{1/3}[/tex]

ardni313 ardni313 · Answer 2 · 2019-07-27T15:47:19+02:00

The radius of a solid aluminum sphere that balances a solid iron sphere of radius rFe on an equal-arm balance

[tex]r_{Al}~=~r_{Fe}\sqrt[3]{\frac{\rho.Fe}{\rho.Al} } }[/tex]

Further explanation

Density is the amount derived from mass and volume

density is the ratio of mass per unit volume

With the same mass, the volume of objects that have a high density will be smaller than objects with a smaller density

The unit of density can be expressed in g / cm³ or kg / m³

Density formula:

[tex]\large{\boxed{\bold{\rho~=~\frac{m}{V} }}}[/tex]

ρ = density

m = mass

v = volume

A common example is the density of water 1 gr / cm³

the mass itself can be measured by the balance sheet, the arm balance

Known density of Al (aluminum) and Fe (iron)

Because both are spherical, the volume = [tex]\frac{4}{3} .\pi .r^3[/tex]

the mass of the two metals is the same because the scales are equal

mass Al = mass of Fe

ρ. V Al = ρ.V Fe

[tex]\rho.V_{Al}~=~\rho.V_{Fe}[/tex]

[tex]\rho_{Al}.\frac{4}{3} \pi.r^3~=~\rho_{Fe}.\frac{4}{3} .r^3[/tex]

[tex]r^3_{Al}~=~\frac{\rho_{Fe}.\frac{4}{3}.\pi.r_{Fe}^3 }{\rho_{Al}.\frac{4}{3} \pi }[/tex]

[tex]r_{Al}~=~r_{Fe}\sqrt[3]{\frac{\rho.Fe}{\rho.Al} } }[/tex]

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Keywords: density, volume, mass, arm balance