In a horizontal pipeline of uniform cross section the pressure falls by `8 N//m^(2)` between two points separated by `1 km`. If oil of density `800 kg//m^(3)` flows through the pipe, find the change in `KE` per kg of oil at these points.

Options

A. `10^(-2)J//kg`
B. `10^(-3)J//kg`
C. `10^(-4)J//kg`
D. `10^(-1)J//kg`

Explanation

To solve the problem, we will use Bernoulli's principle, which relates the pressure, velocity, and height of a fluid flowing in a horizontal pipe. Let's break down the steps:

### Step 1: Understand the given data
- Pressure drop (ΔP) = 8 N/m²
- Density of oil (ρ) = 800 kg/m³
- Distance between the two points (L) = 1 km = 1000 m

### Step 2: Apply Bernoulli's equation
For a horizontal pipe, Bernoulli's equation can be simplified to:
\[ P_1 + \frac{1}{2} \rho V_1^2 = P_2 + \frac{1}{2} \rho V_2^2 \]

Rearranging this gives us:
\[ P_1 - P_2 = \frac{1}{2} \rho V_2^2 - \frac{1}{2} \rho V_1^2 \]

### Step 3: Express the change in kinetic energy per unit mass
The change in kinetic energy per unit mass (ΔKE) can be expressed as:
\[ \Delta KE = \frac{1}{2} \rho V_2^2 - \frac{1}{2} \rho V_1^2 = \frac{P_1 - P_2}{\rho} \]

### Step 4: Substitute the known values
We know:
- \( P_1 - P_2 = 8 \, \text{N/m}^2 \)
- \( \rho = 800 \, \text{kg/m}^3 \)

Substituting these values into the equation gives:
\[ \Delta KE = \frac{8}{800} \]

### Step 5: Calculate ΔKE
Now, perform the calculation:
\[ \Delta KE = \frac{8}{800} = 0.01 \, \text{J/kg} \]

### Final Answer
The change in kinetic energy per kg of oil at these points is:
\[ \Delta KE = 0.01 \, \text{J/kg} \]

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