Determine the percentage change in the volume of water if its pressure is increased by 30,000 psi.
Understand the Problem
The question is asking to determine the percentage change in the volume of water when its pressure is increased by 30,000 psi using the bulk modulus of elasticity of water.
Answer
The percentage change in volume is $-10\%$.
Answer for screen readers
The percentage change in the volume of water when its pressure is increased by 30,000 psi is $-10%$.
Steps to Solve
- Identify the given values
We know the following values:
- Bulk modulus of elasticity for water, $ \beta = 300,000 \text{ psi} $
- Change in pressure, $ \Delta p = 30,000 \text{ psi} $
- Use the formula for percentage change in volume
The formula for the percentage change in volume is given by:
$$ \frac{\Delta V}{V} = -\frac{\Delta p}{\beta} $$
- Substitute the known values into the formula
Now we can substitute the values of $\Delta p$ and $\beta$ into the formula:
$$ \frac{\Delta V}{V} = -\frac{30,000}{300,000} $$
- Calculate the value
Perform the calculation:
$$ \frac{\Delta V}{V} = -0.1 $$
- Convert to percentage
To find the percentage decrease, convert the fraction to a percentage by multiplying by 100:
$$ \text{Percentage change} = -0.1 \times 100 = -10% $$
The percentage change in the volume of water when its pressure is increased by 30,000 psi is $-10%$.
More Information
The negative sign indicates a decrease in volume, which is characteristic of many fluids when subjected to increased pressure. In this case, water's volume decreases by 10% due to the applied pressure.
Tips
- Confusing the bulk modulus with compressibility. The bulk modulus specifically relates to volume change under pressure.
- Forgetting to apply the negative sign to indicate a decrease in volume.
