Question

The diameters of the neck and the bottom of a glass bottle are $2\text{cm}$ and $8\text{cm}$ respectively. The bottle is completely filled with olive oil. If the cork in the bottle's neck is pressed in with a force of $1.4\text{kgf}$, calculate the force exerted on the bottom in $\text{kgf}$.

• A. 22.4

Answer 1

The correct option is A 22.4
Given:
The diameter of the neck, ${\text{d}}_1=2\text{cm}$
The diameter of the bottle, ${\text{d}}_2=8\text{cm}$
Area of a circle $\propto {\text{diameter}}^2$
$⟹{\text{A}}_1\propto {\text{d}}_1^2$ and ${\text{A}}_2\propto {\text{d}}_2^2$, where ${\text{A}}_1$ and ${\text{A}}_2$ are the areas of cross section of the neck and the bottom respectively.
The force applied on the neck, ${\text{F}}_1=1.4\text{kgf}$
If ${\text{F}}_2$ is the resultant force exerted on the bottom, then from Pascal's law:
$\frac{{\text{F}}_2}{{\text{F}}_1}=\frac{{\text{A}}_2}{{\text{A}}_1}$, when ${\text{A}}_1<{\text{A}}_2$
$⟹\frac{{\text{F}}_2}{{\text{F}}_1}=\frac{{\text{d}}_2^2}{{\text{d}}_1^2}$
$⟹{\text{F}}_2={\text{F}}_1\frac{{\text{d}}_2^2}{{\text{d}}_1^2}=1.4\times \frac{8^2}{2^2}$
$⟹{\text{F}}_2=1.4\times 16=22.4\text{kgf}$