Problem Statement
A 10mm ball of relative density 1.2 is suspended from a string, in air flowing at a velocity of 10m/s. Determine the angle which the string will make with the vertical. Take ρ of air = 1.208 kg/m3 and viscosity of air = 1.85×10-5 Pa-s. Also compute the tension in the string.
Given Data
Solution Approach
To solve this problem, we need to analyze the forces acting on the suspended ball. The ball experiences a drag force due to the air flow and a gravitational force due to its weight. These forces will determine the angle of the string with the vertical and the tension in the string.
Calculations
Basic Parameters Calculation
Step 1: Calculate the cross-sectional area of the ball.
Step 2: Calculate the weight of the ball.
For a sphere with relative density 1.2:
Note: γ = relative density × density of water × g = 1.2 × 1000 × 9.81 = 11772 N/m3
Step 3: Calculate the Reynolds number to determine the flow regime.
Since Re falls between 1000 and 100,000, we can use CD = 0.5 for a sphere in this flow regime.
Step 4: Calculate the drag force on the ball.
Step 5: Calculate the angle the string makes with the vertical.
The horizontal force component (drag force) and the vertical force component (weight) determine the angle:
Step 6: Calculate the tension in the string.
The tension can be calculated using the Pythagorean theorem with the horizontal and vertical force components:
Angle with vertical (θ) = 21°
Tension in the string (T) = 0.0066 N
Detailed Explanation
Physical Interpretation
This problem illustrates the balance of forces on an object in a flowing fluid. When the ball is suspended in the flowing air:
- The drag force acts horizontally in the direction of the air flow
- The weight acts vertically downward
- The tension in the string acts along the string direction
At equilibrium, these forces must balance, resulting in the string making an angle with the vertical.
Drag Coefficient Determination
The drag coefficient (CD) is a dimensionless quantity that characterizes the drag or resistance of an object in a fluid environment. For a sphere, CD varies with the Reynolds number:
- Re < 1: CD ≈ 24/Re (Stokes’ Law region)
- 1 < Re < 1000: Transitional region
- 1000 < Re < 100,000: CD ≈ 0.5 (Newton’s Law region)
- Re > 100,000: CD drops due to boundary layer transition (drag crisis)
In our case, Re = 6530, so using CD = 0.5 is appropriate.
Force Balance Analysis
The angle of 21° represents the deflection of the string from the vertical due to the horizontal drag force. This can be understood through a force balance:
- Horizontal balance: T sin θ = FD
- Vertical balance: T cos θ = W
Dividing these equations: tan θ = FD/W = 0.3848, which gives θ = 21°
Applications in Engineering
This type of analysis is important in various engineering applications:
- Design of weather instruments (anemometers)
- Analysis of cable-stayed structures in wind
- Performance of moored vessels or floating platforms
- Aerodynamic loading on suspended objects
- Design considerations for outdoor installations (signs, lights, wires)
Scaling Considerations
It’s worth noting how this scenario would change with different parameters:
- Increasing air velocity would increase the drag force proportionally to V2, resulting in a larger deflection angle
- A larger ball would experience more drag (proportional to d2) but also more weight (proportional to d3). Since weight increases faster than drag with size, larger spheres would deflect less for the same wind speed
- A ball with higher relative density would have greater weight and therefore deflect less
This problem demonstrates the practical application of fluid mechanics principles to predict the behavior of objects in flowing fluids, incorporating concepts of drag forces, Reynolds number, and static equilibrium.
