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%. Calculate the force exerted by the juice per square centimeter if its bulk modulus is 1.8×10^9 N/m^2, assuming the bottle does not break.

a) 0.36 N/cm^2
b) 0.72 N/cm^2
c) 1.08 N/cm^2
d) 1.44 N/cm^2

Answer 1

The force exerted by the juice per square centimeter is 0.36 N/cm².

Step-by-step explanation:

To calculate the force exerted by the juice per square centimeter, we need to use the formula for bulk modulus:

B = (F/A) / [(∆V/V₁)]

Where B is the bulk modulus, F is the force exerted, A is the area, ∆V is the change in volume, and V₁ is the initial volume.

In this case, we are given that ∆V/V₁ = 0.002 (0.2%) and the bulk modulus is 1.8 × 10⁹ N/m².

Plugging in the values, we have:

(F/A) / (0.002) = 1.8 × 10⁹

Now, we need to solve for F/A:

F/A = (1.8 × 10⁹) × (0.002)

Simplifying the equation gives us:

F/A = 3.6 × 10⁶ N/m²

Converting N/m² to N/cm² by dividing by 10,000, we get:

F/A = 0.36 N/cm²

Therefore, the force exerted by the juice per square centimeter is 0.36 N/cm².