Question

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, frac{Δ V}{V₀} = 2 × 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 × 10⁹ , {N/m}², assuming the bottle does not break. In view of your answer, do you think the bottle will survive?

A. 4.36 × 10⁷ , {N/m}², Yes
B. 6.18 × 10⁷ , {N/m}², No
C. 7.95 × 10⁷ , {N/m}², Yes
D. 9.23 × 10⁷ , {N/m}², No

Answer 1

The magnitude of the normal force exerted by the juice per square centimeter is approximately 3.6 × 10⁷ N/cm². The bottle will survive.

Step-by-step explanation:

The magnitude of the normal force exerted by the juice per square centimeter can be calculated using the formula:

F = B × (ΔV/V₀)

Where F is the force exerted, B is the bulk modulus, and ΔV/V₀ is the fractional change in volume. Substituting the given values, we have:

F = (1.8 × 10⁹ N/m²) × (2 × 10⁻³)

F = 3.6 × 10⁶ N/m²

This is approximately equal to 3.6 × 10⁷ N/cm².

Based on the calculated magnitude of the normal force, we can conclude that the bottle will survive, as the force exerted by the juice per square centimeter is within a safe range.