Question

The pressure inside a champagne bottle can be quite high and can launch a cork explosively. Suppose you open a bottle at sea level. The absolute pressure inside a champagne bottle is 6 times atmospheric pressure; the cork has a mass of 7.5 g and a diameter of 18 mm.Assume that once the cork starts to move, the only force that matters is the pressure force.

What is the acceleration of the cork? Express your answer with the appropriate units.

Answer 1

The diameter of the cork is \(18 \mathrm{~mm}\). Thus, the

radius of the cork is, \(r=\frac{18 \mathrm{~mm}}{2}=9 \mathrm{~mm}=9 \times 10^{-3} \mathrm{~m}\)

The force is cal culated as follows:

\(\begin{aligned} F &=\Delta p A \\ &=\Delta p\left(\pi r^{2}\right) \\ &=(6 a t m-1 a t m)\left(\frac{1.01325 \times 10^{5} P a}{1 a t m}\right)\left[\pi\left(9 \times 10^{-3} \mathrm{~m}\right)^{2}\right] \\ & \approx 128.9203596 \mathrm{~N} \end{aligned}\)

The acceleration of the rock is,

\(a=\frac{F}{m}\)

\(\begin{aligned} a &=\frac{128.9203596 \mathrm{~N}}{7.5 \times 10^{-3} \mathrm{~kg}} \\ a &=17189.38 \mathrm{~m} / \mathrm{s}^{2} \end{aligned}\)