Subtopic: Linear Momentum Equation & Jet Impingement
Question Statement:
Water exits from a nozzle of diameter 50 mm into atmospheric pressure at a velocity of 20 m/s. The jet strikes a flat vertical plate held perpendicular to the jet. If the density of water is 1000 kg/m$^3$, the horizontal force (in N) required to hold the plate stationary is _________ (round off to the nearest integer).
Official Answer Key:
785.0
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2017 Q.36)**
**1. Intuitive Physical Setup & Given Parameters**:
- Water jet diameter: $d = 50\text{ mm} = 0.05\text{ m}$.
- Jet exit velocity: $V = 20\text{ m/s}$.
- Density of water: $\rho = 1000\text{ kg/m}^3$.
- Target: Force exerted on the flat stationary normal plate.
**2. Step-by-Step Fluid Mechanics Derivation**:
1. **Jet Cross-Sectional Area ($A$)**:
$A = \frac{\pi d^2}{4} = \frac{\pi (0.05)^2}{4} = \frac{0.0025\pi}{4} \approx 1.9635 \times 10^{-3}\text{ m}^2$
2. **Mass Flow Rate ($\dot{m}$)**:
$\dot{m} = \rho A V = 1000 \times (1.9635 \times 10^{-3}) \times 20 = 39.27\text{ kg/s}$
3. **Control Volume Linear Momentum Balance**:
The fluid strikes the plate normally and deflects radially along the plate surface at $90^\circ$ (exit velocity normal to plate is zero):
$F_x = \dot{m}(V_{in, x} - V_{out, x}) = \dot{m}(V - 0) = \rho A V^2$
4. **Evaluating Impact Force**:
$F_x = 1000 \times (1.9635 \times 10^{-3}) \times (20)^2 = 1.9635 \times 400 = 785.4\text{ N}$
Rounding to the nearest integer:
$F_x \approx 785\text{ N}$
Acceptable official range: **[780, 790] N**.
**3. Comprehensive Options & Distractor Analysis**:
- **Question Type**: Numerical Answer Type (NAT).
- **Official Range**: **780 to 790 N**.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **785.0** (Acceptable Range: **780 to 790**)
- **Key Formula**: $F = \rho A V^2 = \dot{m} V$ for normal stationary plate.
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.