Question

The neck and the bottom of an empty glass bottle are $1.6\text{cm}$ and $7.2\text{cm}$ in diameter respectively. A cork is pressed into the mouth of the bottle with a uniform thrust of $1.2\text{kgf}$. Find the force exerted on the bottom of the bottle in $\text{N}$.[$5\text{marks}$]

Answer 1

Given:
The diameter of the bottle's neck $=1.6\text{cm}$
$⟹$ Radius of the bottle's neck, ${\text{r}}_1=0.8\text{cm}$
$⟹$ Area of the bottle's neck, ${\text{A}}_{\text{neck}}=\pi (0.8{)}^2=0.64\pi {\text{cm}}^2$
The diameter of the bottle's bottom $=7.2\text{cm}$
$⟹$ Radius of the bottle's bottom, ${\text{r}}_2=3.6\text{cm}$
$⟹$ Area of the bottle's neck, ${\text{A}}_{\text{bottom}}=\pi (3.6{)}^2=12.96\pi {\text{cm}}^2$
Assuming the cork has the same diameter of cross section as the bottle's neck, the thrust applied by the cork, $\text{F}=1.2\text{kgf}$ [$2\text{marks}$] Since it is the same bottle all the way down, the pressure experienced at the neck due to the plugging of the cork should be the same as the pressure felt at the bottom.
$⟹\frac{{\text{F}}_{\text{neck}}}{{\text{A}}_{\text{neck}}}=\frac{{\text{F}}_{\text{bottom}}}{{\text{A}}_{\text{bottom}}}$
$⟹{\text{F}}_{\text{bottom}}=\frac{{\text{F}}_{\text{neck}}}{{\text{A}}_{\text{neck}}}\times {\text{A}}_{\text{bottom}}$
$⟹{\text{F}}_{\text{bottom}}=\frac{1.2\text{kgf}}{0.64\pi {\text{cm}}^2}\times 12.96\pi {\text{cm}}^2$

$⟹{\text{F}}_{\text{bottom}}=1.2\times 20.25=24.3\text{kgf}$

$1\text{kgf}=9.807\text{N}\approx 10\text{N}$

$\therefore {\text{F}}_{\text{bottom}}=24.3\times 10=243\text{N}$ [$3\text{marks}$]