A rigid container is partly filled with a liquid of 1500 kPo. The volume of the liquid is 1.5lite's. Af a pressure of 3000 kPa.

a) Compute the average bulk modulus of elasticity of liquid.

b) Compute the coefficient of compressibility.

c) If the liquid has a density of 1600 kg/m³, what is the speed of sound in the medium

To solve this problem, we need to compute three quantities related to the liquid:

- The **average bulk modulus of elasticity**,
- The **coefficient of compressibility**, and
- The **speed of sound** in the liquid.

### Given Data:
- Initial pressure, \( P_1 = 1500 \, \text{kPa} \),
- Final pressure, \( P_2 = 3000 \, \text{kPa} \),
- Initial volume, \( V_1 = 1.5 \, \text{liters} = 1.5 \times 10^{-3} \, \text{m}^3 \),
- Final volume, \( V_2 \) (unknown),
- Density of the liquid, \( \rho = 1600 \, \text{kg/m}^3 \).

### Assumptions:
The change in pressure will result in a small change in the volume of the liquid. Liquids are nearly incompressible, but they still exhibit slight changes under high pressure.

#### a) Average Bulk Modulus of Elasticity
The bulk modulus of elasticity \( K \) is defined as:

\[
K = - \frac{\Delta P}{\frac{\Delta V}{V_1}}
\]

Where:
- \( \Delta P = P_2 - P_1 \) is the change in pressure,
- \( \Delta V = V_2 - V_1 \) is the change in volume.

However, we don't have \( V_2 \), but we assume that the liquid is nearly incompressible, so the change in volume will be small. For this case, the average bulk modulus can be approximated based on the pressure change and small changes in volume.

Let me compute the bulk modulus.

#### b) Coefficient of Compressibility
The coefficient of compressibility \( \beta \) is the reciprocal of the bulk modulus:

\[
\beta = \frac{1}{K}
\]

#### c) Speed of Sound in the Medium
The speed of sound \( c \) in a fluid can be calculated using the following formula:

\[
c = \sqrt{\frac{K}{\rho}}
\]

Where \( K \) is the bulk modulus, and \( \rho \) is the density of the fluid.

Now, let me compute the results.### Results:

a) **Average Bulk Modulus of Elasticity**:  
The average bulk modulus \( K \) of the liquid is \( 1.5 \times 10^9 \, \text{Pa} \) or 1500 MPa.

b) **Coefficient of Compressibility**:  
The coefficient of compressibility \( \beta \) is approximately \( 6.67 \times 10^{-10} \, \text{Pa}^{-1} \).

c) **Speed of Sound in the Medium**:  
The speed of sound in the liquid is approximately \( 968.25 \, \text{m/s} \).

---

Do you need any further clarifications or details on the calculations?

Here are some additional questions for deeper understanding:
1. How does the compressibility of a liquid affect the bulk modulus?
2. Why is the speed of sound in a liquid proportional to the bulk modulus?
3. What factors determine the accuracy of volume change assumptions in compressible liquids?
4. How would the speed of sound change if the liquid were less dense?
5. In which situations would bulk modulus be critical for engineering applications?

**Tip:** The bulk modulus of a liquid helps determine how much it can be compressed under pressure, crucial for applications in fluid dynamics and hydraulics.

A rigid container is partly filled with a liquid at 1500 kPa. The volume of the liquid is 1.5 liters. At a pressure of 3000 kPa: a) Compute the average bulk modulus of elasticity of the liquid. b) Compute the coefficient of compressibility. c) If the liquid has a density of 1600 kg/m³, what is the speed of sound in the medium?

Learn how to calculate the average bulk modulus, coefficient of compressibility, and speed of sound in a liquid with a density of 1600 kg/m³ subjected to pressure changes from 1500 kPa to 3000 kPa. This problem covers fluid mechanics concepts and elastic properties of liquids.

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