The pressure increases by 1.0 x 104 N/m2 for every meter of depth beneath the surface of the ocean. At what depth does the volume of a Pyrex (bulk modulus 2.6 x 1010 N/m2) glass cube, 2.9 x 10-2 m on an edge at the ocean's surface, decrease by 9.5 x 10-10 m3?
pretty deep
To solve this problem, we need to use the concept of pressure and the bulk modulus of the material.
The change in volume (∆V) of the cube is related to the change in pressure (∆P) and the bulk modulus (K) through the formula:
∆V = -V₀ (∆P / K)
Where:
- ∆V is the change in volume of the cube
- V₀ is the initial volume of the cube
- ∆P is the change in pressure
- K is the bulk modulus
From the given information:
- ∆V = -9.5 x 10^-10 m^3
- V₀ = (2.9 x 10^-2 m)^3
- K = 2.6 x 10^10 N/m^2
We know that the pressure increases by 1.0 x 10^4 N/m^2 for every meter of depth. Therefore, the change in pressure (∆P) can be calculated as follows:
∆P = (1.0 x 10^4 N/m^2) × (depth)
Now, we can substitute these values into the formula and solve for the depth (depth = d):
-9.5 x 10^-10 m^3 = -(2.9 x 10^-2 m)^3 (1.0 x 10^4 N/m^2) × d / (2.6 x 10^10 N/m^2)
Simplifying the equation:
-9.5 x 10^-10 m^3 = -0.024389 x 10^-6 m^3 × d / (2.6 x 10^10)
Cross-multiplying and rearranging the equation:
d = (-9.5 x 10^-10 m^3) × (2.6 x 10^10 N/m^2) / (-0.024389 x 10^-6 m^3)
d ≈ 3.9 meters
Therefore, the volume of the Pyrex glass cube decreases by 9.5 x 10^-10 m^3 when the depth beneath the surface of the ocean is approximately 3.9 meters.