Question

The weight of a metallic solid sphere varies as the density of its material when its diameter is constant and varies as the diameter when density is constant. A sphere having density 2.1gm/cm³ and diameter 2 cm, weighs 2.2 gm. Find the weight of a sphere whose diameter is 1 cm and density is 6.3 gm/cm³.

Answer 1

$$\begin{gathered} \mathrm{Let}W,D\mathrm{and}d\mathrm{denote}\mathrm{the}\mathrm{weight},\mathrm{density}\mathrm{and}\mathrm{diameter}\mathrm{of}\mathrm{the}\mathrm{metallic}\mathrm{solid}\mathrm{sphere},\mathrm{respectively}. \\ \mathrm{Given}: \\ W\alpha D...\left(\mathrm{i}\right)\left(d\mathrm{is}\mathrm{a}\mathrm{constant}.\right) \\ W\alpha d...\left(ii\right)\left(D\mathrm{is}\mathrm{a}\mathrm{constant}.\right) \\ \mathrm{From}\left(\mathrm{i}\right)\&\left(\mathrm{ii}\right),\mathrm{we}\mathrm{get}: \\ W\alpha Dd \\ \Rightarrow W=kDd\left(k\mathrm{is}a\mathrm{constant}.\right) \\ Wehave: \\ W=2.2\mathrm{gm}\mathrm{when}D=2.1\mathrm{gm}/{\mathrm{cm}}^{3}andd=2\mathrm{cm}. \\ \Rightarrow 2.2=k\times 2.1\times 2 \\ \Rightarrow k=\frac{2.2}{2.1\times 2} \\ \Rightarrow k=0.52 \\ \mathrm{When}d=1\mathrm{cm}\mathrm{and}D=6.3\mathrm{gm}/{\mathrm{cm}}^3,\mathrm{we}\mathrm{get}: \\ W=0.52\times 6.3\times 1 \\ =3.27\mathrm{gm}\approx 3.3\mathrm{gm} \end{gathered}$$