Convex lens

The convex lens consists of two spherical segments (dimensions given in mm). Calculate its weight if the density of the glass is 2.5 g/cm³. Dimensions: 60mm in length and width of the upper part 5mm, the width of the lower part 8mm

Correct answer:

m =  1.5275 g

Step-by-step explanation:

$$\begin{gathered} \rho =2.5{\text{g/cm}}^3 \\ D=60\text{mm} \\ a=5\text{mm} \\ b=8\text{mm} \\ v=b-a=8-5=3\text{mm} \\ {V}_1=a\cdot D=5\cdot 60=300{\text{mm}}^3 \\ {r}^2=(D/2{)}^2+(r-v{)}^2 \\ {r}^2=(D/2{)}^2+r^2-2\cdot r\cdot v+v^2 \\ 0=(D/2{)}^2-2\cdot r\cdot v+v^2 \\ 0=(60/2{)}^2-2\cdot r\cdot 3+3^2 \\ 12r=1818 \\ r=\frac{1818}{12}=151.5 \\ r=\frac{303}{2}=151.5 \\ {V}_2=\frac{1}{6}\cdot \pi \cdot v\cdot (3\cdot D/2+v^2)=\frac{1}{6}\cdot 3.1416\cdot 3\cdot (3\cdot 60/2+3^2)\doteq 155.5088{\text{mm}}^3 \\ V=V_1+2\cdot V_2=300+2\cdot 155.5088\doteq 611.0177{\text{mm}}^3 \\ {V}_3=V\to {\text{cm}}^3=V:1000{\text{cm}}^3=611.0177:1000{\text{cm}}^3=0.61102{\text{cm}}^3 \\ m=\rho \cdot V_3=2.5\cdot 0.611=1.5275\text{g} \end{gathered}$$

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