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, 5.6 x 10-2 m on an edge at the ocean's surface, decrease by 3.0 x 10-10 m3?
Why did the Pyrex glass cube visit the ocean?
Because it wanted to experience the deep-end pressure!
Okay, let's calculate the depth at which the volume of the Pyrex glass cube decreases.
The change in volume (∆V) can be calculated using the formula:
∆V = -V0 * ∆P / B
Where:
- V0 is the initial volume of the glass cube at the ocean's surface.
- ∆P is the change in pressure.
- B is the bulk modulus of Pyrex glass.
Given:
V0 = (5.6 x 10^-2 m)^3 (volume of the cube at the ocean's surface)
∆P = 1.0 x 10^4 N/m^2 (pressure per meter of depth)
B = 2.6 x 10^10 N/m^2 (bulk modulus of Pyrex glass)
∆V = -3.0 x 10^-10 m^3 (change in volume)
Let's put these values into the formula. But be careful, I'll do the math, so don't worry about it!
∆V = -V0 * ∆P / B
-3.0 x 10^-10 m^3 = -(5.6 x 10^-2 m)^3 * (1.0 x 10^4 N/m^2) / (2.6 x 10^10 N/m^2)
And the result is...
*dramatic drumroll*
The depth at which the volume of the Pyrex glass cube decreases by 3.0 x 10^-10 m^3 is approximately 20.1 meters beneath the ocean's surface.
Now, if the glass cube starts singing "Under the Sea," you'll know why!
To find the depth at which the volume of the Pyrex glass cube decreases by a certain amount, we need to use the equation relating pressure and depth, as well as the equation for bulk modulus and volume change.
Step 1: Convert the given dimensions into proper units.
The edge length of the glass cube is given as 5.6 x 10^-2 m.
Step 2: Calculate the initial volume of the glass cube.
The initial volume (V_0) of the glass cube can be found by using the formula for the volume of a cube: V_0 = (edge length)^3.
V_0 = (5.6 x 10^-2 m)^3
Step 3: Calculate the change in volume.
The change in volume (ΔV) is given as 3.0 x 10^-10 m^3.
ΔV = 3.0 x 10^-10 m^3
Step 4: Calculate the bulk modulus.
The bulk modulus (B) of Pyrex glass is given as 2.6 x 10^10 N/m^2.
B = 2.6 x 10^10 N/m^2
Step 5: Calculate the change in pressure.
The change in pressure (ΔP) can be calculated by rearranging the formula for bulk modulus:
ΔP = B * (ΔV / V_0)
Step 6: Solve for depth.
The change in pressure (ΔP) is given by the formula:
ΔP = ρ * g * Δh
Where:
ρ = density of the fluid (assume water) = 1000 kg/m^3
g = acceleration due to gravity = 9.8 m/s^2
Δh = change in depth
By equating the expressions for ΔP, we can solve for Δh:
B * (ΔV / V_0) = ρ * g * Δh
Δh = (B * ΔV) / (ρ * g * V_0)
Step 7: Substitute the given values and calculate.
Substituting the given values into the equation, we can calculate the depth:
Δh = ((2.6 x 10^10 N/m^2) * (3.0 x 10^-10 m^3)) / ((1000 kg/m^3) * (9.8 m/s^2) * (5.6 x 10^-2 m)^3)
To find the depth at which the volume of the Pyrex glass cube decreases by a given value, we need to use the equation of bulk modulus:
ΔV/V = -B(ΔP / P)
Where:
ΔV = Change in volume
V = Initial volume
B = Bulk modulus
ΔP = Change in pressure
P = Initial pressure
In this case, the initial pressure at the surface of the ocean is 1.0 x 10^4 N/m^2, and the change in pressure can be calculated from the change in volume using the given values:
ΔP = (ΔV / V) * P
Plugging in the values, we have:
ΔP = (3.0 x 10^-10 m^3 / (5.6 x 10^-2 m)^3) * (1.0 x 10^4 N/m^2)
Now, we can rearrange the equation to find the depth, which is the change in pressure divided by the pressure change per depth:
depth = ΔP / (1.0 x 10^4 N/m^2)
Calculating the values:
depth = [(3.0 x 10^-10 m^3 / (5.6 x 10^-2 m)^3) * (1.0 x 10^4 N/m^2)] / (1.0 x 10^4 N/m^2)
Simplifying further:
depth = 3.0 x 10^-10 m^3 / (5.6 x 10^-2 m)^3
depth = 3.0 x 10^-10 m^3 / 1.08656 x 10^-5 m
depth ≈ 2.759 x 10^-5 m
Therefore, the volume of the Pyrex glass cube decreases by 3.0 x 10^-10 m^3 at a depth of approximately 2.759 x 10^-5 meters below the surface of the ocean.