To determine the speed at which air must flow over the upper surface of a square plate to keep it horizontal, we can apply the principle of dynamic lift. This principle states that an increase in the speed of a fluid over a surface results in a decrease in pressure on that surface, which can create lift. In this case, we want the lift generated by the airflow to equal the weight of the plate.
Understanding the Forces Involved
The plate has a mass of 488 grams, which we can convert to kilograms for consistency in our calculations:
- Mass of the plate (m) = 488 g = 0.488 kg
The weight (W) of the plate can be calculated using the formula:
Where g is the acceleration due to gravity (approximately 9.81 m/s²). Thus:
- W = 0.488 kg × 9.81 m/s² = 4.78568 N
Calculating the Lift Force
The lift force (L) generated by the airflow can be expressed using Bernoulli's principle, which relates the pressure difference to the velocity of the fluid. The lift force can be calculated as:
Where:
- P₁ is the atmospheric pressure (approximately 101,325 Pa),
- P₂ is the pressure above the plate, and
- A is the area of the plate.
The area (A) of the square plate is given by:
- A = edge length × edge length = 0.0910 m × 0.0910 m = 0.008273 m²
Applying Bernoulli's Equation
According to Bernoulli's equation, the pressure difference can be related to the velocity of the air (v) flowing over the plate:
Where ρ is the density of the air (1.21 kg/m³). Setting the lift force equal to the weight of the plate gives us:
- 4.78568 N = (0.5 × 1.21 kg/m³ × v²) × 0.008273 m²
Solving for Velocity
Now we can rearrange the equation to solve for v:
- 4.78568 N = 0.5 × 1.21 × 0.008273 × v²
- 4.78568 = 0.0050065 × v²
- v² = 4.78568 / 0.0050065
- v² ≈ 955.51
- v ≈ √955.51 ≈ 30.93 m/s
Final Result
To keep the square plate horizontal, the air must flow over its upper surface at a speed of approximately 30.93 m/s. This speed creates enough lift to counteract the weight of the plate, allowing it to remain suspended in a horizontal position.
