Problem Statement
Solve the fluid mechanics problem: Show that the capillary rise height is inversely proportional to the tube radius (Jurin’s law). Calculate $h$ for water in capillaries of $r=0.1\ \text{mm}$ and $r=1.0\ \text{mm}$. $$h = \frac{2\sigma\cos\theta}{\rho g r} \propto \frac{1}{r}$$ This is Jurin’s law (1718): narrower tubes pull liquid h
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
Surface tension $\sigma$ is the energy per unit area (or force per unit length) at a liquid surface. It arises from cohesive intermolecular forces. Capillary rise results from the balance between surface tension pulling liquid up and gravity pulling it down. The Laplace pressure across a curved interface is $\Delta P = 2\sigma/r$ (sphere) or $\sigma/r$ (cylinder).
- $h = 2\sigma\cos\theta/(\rho g r)$ — capillary height
- $\Delta P = 2\sigma/r$ — excess pressure inside a droplet
- $W = \sigma \Delta A$ — work done against surface tension
Step-by-Step Solution
Step 1 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 2 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Worked Calculation
$$h = \frac{2\sigma\cos\theta}{\rho g r} \propto \frac{1}{r}$$
$$P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g h_2$$
$$v = \sqrt{2gh} = \sqrt{2\times9.8\times2} = \sqrt{39.2} \approx 6.26\,\text{m/s}$$
Answer
$$\boxed{v_{\text{efflux}} = \sqrt{2gh}}$$
Physical Interpretation
Capillary action allows plants to draw water from roots to leaves against gravity. The thinner the tube, the higher the rise — but also the smaller the volume transported.
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