Density = Mass / Volume
Hence, Mass = Volume \(\times\) Density
= 125 \(\times\) 7.8 = 975 g.

One light year is defined as the distance travelled by light in vacuum during the period of one year.
1 Light year = 9.46 \(\times\) 10¹⁵m

Density of a substance is defined as the mass of the substance contained in unit volume (1 m³).
Density (D) = mass (M)/volume(V)
D = M/V
SI unit of density - kg/m³

i) Find the ratio of masses of spheres A and B.
Sphere A and B are made of the same material
\(\therefore\) The Density of A and B are equal. Take it as D
\(D =\frac{M}{V} \)
\(\therefore M =D \times V\)
Volume of sphere \(A=V_{1}\)
Volume of sphere \(B=V_{2}\)
Mass of sphere \(A=M_{1}\)
Mass of sphere \(\mathrm{B}=\mathrm{M}_{2}\)
\(\therefore \mathrm{M}_{\mathrm{l}} =\mathrm{D} \times \mathrm{V}_{1} \)
\(\mathrm{M}_{2} =\mathrm{D} \times \mathrm{V}_{2}\)
The radius of sphere B is half that of A
So, we can take the radius of sphere A(r)=2
Volume of sphere \(\mathrm{A}\left(\mathrm{V}_{1}\right)=\frac{4}{3} \pi \mathrm{r}^{3}=\frac{4}{3} \times \pi \times 2 \times 2 \times 2\)
Volume of sphere \(A\left(V_{1}\right)=\frac{32 \pi{}}{3}\)
We can take the radius of sphere B(r)=1
Volume of sphere \(B\left(V_{2}\right)=\frac{4}{3} \times \pi \times 1 \times 1 \times 1\)
Volume of sphere \(\mathrm{B}\left(\mathrm{V}_{2}\right)=4 / 3 \neq 0\)
Mass of sphere \(\mathrm{A} \mathrm{M}_{1}=\mathrm{D} \times \frac{32 \pi}{3}\)
Mass of sphere \(\mathrm{B} \mathrm{M}=\mathrm{D} \times \frac{4}{3} \pi\)
Ratio of masses of spheres A and B = \(\frac{\mathrm{M}_{1}}{\mathrm{M}_{2}}\)
\(= \frac{\not D \times \not 32 \times \not  \frac{\pi}{3}}{\not D \times \not  4 \not  \frac{\pi}{3}} \)
\(\mathrm{M}_{1} / \mathrm{M}_{2} =8 / 1 \)
\(\mathrm{M}_{1} \cdot \mathrm{M}_{2} =8: 1\)
Ratio of masses of spheres A and B = 8 : 1
ii) Find the ratio of volume spheres A and B
Volume of sphere \(A\left(V_{1}\right)=\frac{32 \pi}{3}\)
Volume of sphere \(B\left(V_{2}\right)=\frac{4}{2} \pi\)
Ratio of volume of spheres A and B = \(\frac{V_{1}}{V_{2}}\)
\(=\frac{\not  32 \times \not  \frac{\pi}{3}}{\not  4 \not  \frac{\pi}{3}}=8 / 1\)
So Ratio of volume of spheres A and B = 8 : 1
\(V_{1}: V_{2}=8: 1\)
iii) Find the ratio of masses of spheres A and C
i) Spheres A and C have equal radii so the volume of A and C equal and take it as V
ii) Density of A is double that of C now we can take.
Density of A=2 D
Density of C=D
\(\mathrm{M} =\mathrm{D} \times \mathrm{V} \)
\(\text { Mass of sphere } \mathrm{A} =\mathrm{M}_{1} \quad \mathrm{M}_{1}=2 \mathrm{D} \times \mathrm{V} \)
\(\text { Mass of sphere } \mathrm{C} =\mathrm{M}_{3} \quad \mathrm{M}_{3}=\mathrm{D} \times \mathrm{V} \)
\(\mathrm{M}_{1}: \mathrm{M}_{3} =2 \mathrm{D} \times \mathrm{V} / \mathrm{D} \times \mathrm{V}=2 / 1 \)
\(\mathrm{M}_{\mathrm{1}}: \mathrm{M}_{3} =2: \mathrm{1} \)
Ratio of masses of sphere A and C = 2 : 1