Torricelli’s Law and Conservation of Momentum

Abstract

This TFG discusses Conservation of Momentum in the context of Torricelli’s law. We consider a large tank filled with water with a tiny hole near its base, and compute the horizontal force exerted on the fluid by the container walls in two ways, finding that the results differ by a factor of 2. We obtain that 𝐹𝑥 = 2𝜌𝑔ℎ𝑆2, where 𝑆2 is the cross-sectional area of the hole, after applying conservation of momentum, and 𝐹𝑥 = 𝜌𝑔ℎ𝑆2 using Cauchy’s hypothesis. We analyze the assumptions made in these calculations, and perform experiments to confirm that some of them are not realized, identifying the sources of the discrepancy. Specifically, the assumption that the pressure is hydrostatic everywhere in the tank except at the hole and that the velocity is strictly horizontal are the key reasons for our discrepancy. Finally, we discuss Borda’s Mouthpiece and redefine the selected control volume as a way to reconcile both approaches to obtain the force.