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, 6.4 x 10-2 m on an edge at the ocean's surface, decrease by 5.8 x 10-10 m3?
I know that I need to use DeltaP=-B(DeltaV/V initial) but the wording is very weird so I'm having trouble with that.
To solve this problem, we can apply the formula for bulk modulus, which relates the change in pressure (ΔP) to the change in volume (ΔV) relative to the initial volume (V initial) and the bulk modulus (B).
Given:
Bulk modulus (B) of Pyrex glass = 2.6 x 10^10 N/m^2
Change in volume (ΔV) = -5.8 x 10^-10 m^3
Initial volume (V initial) = (6.4 x 10^-2 m)^3
We are asked to find the depth at which the volume of the cube decreases by ΔV.
Step 1: Convert the change in volume to a positive value.
Since the formula uses a positive change in volume, we can rewrite the formula as:
ΔP = -B(ΔV / |V initial|)
Step 2: Calculate the initial pressure difference.
The initial pressure difference ΔP can be calculated by multiplying the change in depth (Δh) by the pressure per unit depth. In this case, the pressure increases by 1.0 x 10^4 N/m^2 per meter of depth.
ΔP = (1.0 x 10^4 N/m^2)(Δh)
Step 3: Rearrange the formula to solve for the change in height (Δh).
Rearranging the formula:
Δh = ΔP / (1.0 x 10^4 N/m^2)
Step 4: Calculate the change in pressure.
Using the formula ΔP = -B(ΔV / |V initial|), we can substitute the known values to find the change in pressure.
ΔP = - (2.6 x 10^10 N/m^2) * (-5.8 x 10^-10 m^3) / (6.4 x 10^-2 m)^3
Simplifying the equation above will yield the value for ΔP.
Step 5: Substitute the calculated ΔP into the equation for Δh.
Using the calculated value of ΔP, substitute it into the equation for Δh to find the change in height.
Δh = ΔP / (1.0 x 10^4 N/m^2)
By calculating the final value of Δh, we can determine the depth at which the volume of the Pyrex glass cube decreases by the specified amount of ΔV (5.8 x 10^-10 m^3).