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, consider the relationship between the flow velocity and the pressure difference to calculate the forces.

Answer 1

The Correct Option is B

Step 1: Understanding Bernoulli's principle.
Bernoulli's principle states that the sum of the pressure and kinetic energy per unit volume is constant along a streamline. The air entering the hole generates an upward force due to the change in pressure, which is related to the velocity of the air. The relationship between the force and the geometry of the setup can be derived by applying Bernoulli's principle and integrating the velocity distribution over the area of the hole.
Step 2: Conclusion.
Thus, the upward force on the lower plate is given by option (B).