Density and Pressure — Question 4

Question 4

The same pressure can produce unequal forces

Two small flat wall patches experience the same uniform fluid pressure $p=60\,\mathrm{kPa}$. Patch A has area $2.0\,\mathrm{cm^2}$ and outward normal $\hat{\mathbf i}$; patch B has area $3.0\,\mathrm{cm^2}$ and outward normal $(3\hat{\mathbf i}+4\hat{\mathbf j})/5$. Compute forces exerted by this fluid only.

1. Find each force vector and their vector sum.

2. Why is neither equal force nor a preferred direction implied by equal pressure?

Question 4 - Solution

The same pressure can produce unequal forces

(a) Derivation and calculation

$$\mathbf F=pA\hat{\mathbf n},\quad A_A=2.0\times10^{-4},\quad A_B=3.0\times10^{-4}\,\mathrm{m^2}.$$$$\boxed{\mathbf F_A=(60000)(2.0\times10^{-4})\hat{\mathbf i}=12\hat{\mathbf i}\,\mathrm N}.$$$$\boxed{\mathbf F_B=18(3\hat{\mathbf i}+4\hat{\mathbf j})/5=(10.8\hat{\mathbf i}+14.4\hat{\mathbf j})\,\mathrm N}.$$

(b) Result and interpretation

$$\boxed{\mathbf F_A+\mathbf F_B=(22.8\hat{\mathbf i}+14.4\hat{\mathbf j})\,\mathrm N}.$$$$F_B/F_A=A_B/A_A=1.50.$$Pressure is scalar. Area fixes force magnitude and the local normal fixes direction; equal pressure implies neither equal force nor a preferred spatial direction.

Physical check

This sum concerns the two specified patches only; external pressure and other vessel walls have not been included.