A farmer making grape juice fills a glass bottle to the brim...

A farmer making grape juice fills a glass bottle to the brim and caps it tightly. The juice expands more than the glass when it warms up, in such a way that the volume increases by $0.2 \%$ (that is, $\Delta V / V_{0}=2 \times 10^{-3}$ ) relative to the space available. Calculate the magnitude of the normal force exerted by the juice per square centimeter if its bulk modulus is $1.8 \times 10^{9} \mathrm{N} / \mathrm{m}^{2},$ assuming the bottle does not break. In view of your answer, do you think the bottle will survive?

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Bulk Modulus

Bulk modulus is a measure of a material's resistance to uniform compression. It is defined as the ratio of the change in pressure applied to the material to the resulting fractional change in volume. In problems involving fluids or liquids, it relates the pressure increase caused by a small increase in volume to the inherent stiffness of the substance.

Volumetric Strain

Volumetric strain refers to the fractional change in volume of a substance when subjected to stress. It is the ratio of the change in volume to the original volume. This concept is crucial when determining how much pressure is generated in a fluid due to its expansion or compression, especially when the container’s volume is fixed.

Pressure and Force per Unit Area

Pressure is defined as the force applied per unit area and is a central concept when analyzing stresses on the walls of a container. In the context of fluids in sealed containers, the expansion of the fluid increases the internal pressure, which exerts a normal force on the container's walls. Converting pressure to a force per specific area unit (like per square centimeter) provides insight into whether the container is likely to withstand the stress.

Transcript

Please give Ace some feedback