Problem Statement
Solve the fluid mechanics problem: Find the depression of mercury in a glass capillary of radius $r=1.0\ \text{mm}$. ($\sigma_{Hg}=0.50\ \text{N/m}$, $\theta=140°$, $\rho_{Hg}=13600\ \text{kg/m}^3$) Mercury does not wet glass ($\theta>90°$, $\cos\theta $$h = \frac{2\sigma\cos\theta}{\rho g r}$$ $\cos140° = -0.766$. $$h = \frac{2\time
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
Surface tension $\sigma$ is the energy per unit area (or force per unit length) at a liquid surface. It arises from cohesive intermolecular forces. Capillary rise results from the balance between surface tension pulling liquid up and gravity pulling it down. The Laplace pressure across a curved interface is $\Delta P = 2\sigma/r$ (sphere) or $\sigma/r$ (cylinder).
- $h = 2\sigma\cos\theta/(\rho g r)$ — capillary height
- $\Delta P = 2\sigma/r$ — excess pressure inside a droplet
- $W = \sigma \Delta A$ — work done against surface tension
Step-by-Step Solution
Step 1 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 2 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Worked Calculation
Full substitution shown in the steps above.
Answer
$$\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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