Question

To demonstrate Bernoulli's principle, an instructor arranges two circular horizontal plates of radii \( b \) each with distance \( d \) (\( d \ll b \)) between them (see figure). The upper plate has a hole of radius \( a \) in the middle. On blowing air at a speed \( v_0 \) through the hole so that the flow rate of air is \( \pi a^2 v_0 \), it is seen that the lower plate does not fall. If the density of air is \( \rho \), the upward force on the lower plate is well approximated by the formula (assume that the region with \( r<a \) does not contribute to the upward force and the speed of air at the edges is negligible): 

• A. \( \pi \rho v_0^2 a^4 / 4d^2 \ln \left( \frac{b}{a} \right) \)
• B. \( \pi \rho v_0^2 a^2 b^2 / 4d^2 \ln \left( \frac{b}{a} \right) \)
• C. \( \pi \rho v_0^2 a^4 / 2ab \ln \left( \frac{b}{a} \right) \)
• D. \( 2\pi \rho v_0^2 a^2 / d^2 \ln \left( \frac{b}{a} \right) \)

Hint:

In Bernoulli’s principle problems involving airflow and pressure differences, always consider how the velocity changes with distance and the resulting effect on pressure.

Answer 1

The Correct Option is A

Step 1: Understanding Bernoulli's principle.
Bernoulli's principle states that for an incompressible, non-viscous flow, the total mechanical energy (pressure, kinetic, and potential) along a streamline is constant. For the system in the problem, air flows through a hole of radius \( a \), with a velocity \( v_0 \). The flow rate of the air through the hole is given by \( Q = \pi a^2 v_0 \), where \( v_0 \) is the velocity of the air. We can use Bernoulli’s equation to understand the dynamics of the air as it passes through the hole and applies force on the lower plate. \[ \text{Pressure} + \frac{1}{2} \rho v^2 = \text{constant}. \] The air passing through the hole moves at a high velocity, creating a drop in pressure in the region below the hole. This pressure difference results in an upward force on the lower plate. Step 2: Relating the upward force to the air velocity.
The velocity of the air as it passes through the hole is \( v_0 \). As the air accelerates through the hole, the pressure in the region below the hole decreases, resulting in an upward force on the lower plate. The upward force can be found by considering the velocity distribution of the air in the region below the hole. In the region \( r>a \), the velocity distribution is influenced by the Bernoulli equation, and the force on the lower plate depends on the change in velocity and the cross-sectional area of the flow. Step 3: Deriving the upward force.
The velocity \( v(r) \) at any point in the region below the hole, due to the Bernoulli principle, can be approximated by a logarithmic dependence on the distance \( r \) from the center of the hole. The force is integrated over the area of the hole, and the resulting upward force on the plate is related to the parameters of the system. The expression for the upward force on the lower plate, after integrating the pressure over the area of the hole, is given by: \[ F_{\text{upward}} = \pi \rho v_0^2 a^4 / 4d^2 \ln \left( \frac{b}{a} \right). \] This equation gives the magnitude of the upward force on the lower plate as a function of the density of air \( \rho \), the velocity of air \( v_0 \), and the geometry of the system. Step 4: Conclusion.
Thus, the correct expression for the upward force on the lower plate is given by option (A): \[ \boxed{F_{\text{upward}} = \pi \rho v_0^2 a^4 / 4d^2 \ln \left( \frac{b}{a} \right)}. \]