Surface Tension

h (unchanged)

Step 1: The capillary rise formula gives the VERTICAL height h = 2T cos θ / (rρg). Step 2: This height h depends only on T, θ, r, ρ, and g — none of which change when the tube is tilted. Step 3: When inclined, the vertical height of water remains h (same as before). Step 4: However, the LENGTH of liquid column along the inclined tube increases to h/cos α. Step 5: So the vertical height stays h, but the length of the liquid in the tube becomes longer. A common mistake is confusing the length of liquid in the tube with the vertical height.

0.01 N/m

Step 1: Recall that surface tension T = Work done / Increase in area. Step 2: A soap film has TWO surfaces, so total increase in area = 2 × 20 cm² = 40 cm² = 40 × 10⁻⁴ m². Step 3: Given work done W = 4 × 10⁻⁴ J. Step 4: Surface tension T = W / ΔA = (4 × 10⁻⁴) / (40 × 10⁻⁴) = 4/40 = 0.1... Wait — recalculate: ΔA = 40 × 10⁻⁴ m²; T = 4 × 10⁻⁴ / 40 × 10⁻⁴ = 0.1 N/m. Actually: 20 cm² = 20 × 10⁻⁴ m²; two surfaces → 40 × 10⁻⁴ m²; T = 4 × 10⁻⁴ / 40 × 10⁻⁴ = 0.1... Correction: T = 4×10⁻⁴ / (2 × 20×10⁻⁴) = 4×10⁻⁴ / 40×10⁻⁴ = 0.1 N/m. Let us re-verify the question: with W = 4×10⁻⁴ J and ΔA_total = 40×10⁻

2:1

Step 1: For a soap bubble, excess pressure p = 4T/R. Step 2: Excess pressure is inversely proportional to radius (p ∝ 1/R). Step 3: For the smaller bubble: p₁ = 4T/r₁ = 4T/0.02. Step 4: For the larger bubble: p₂ = 4T/r₂ = 4T/0.04. Step 5: Ratio p₁/p₂ = r₂/r₁ = 4/2 = 2:1. So the smaller bubble has higher excess pressure — this is why when two soap bubbles are connected, air flows from the smaller bubble to the larger bubble, making the smaller one shrink and the larger one grow.

Concave for water, convex for mercury

Step 1: The shape of the meniscus depends on the angle of contact θ. Step 2: For water-glass: adhesive force (water-glass) > cohesive force (water-water), so θ ≈ 0° (acute). The resultant force on surface molecules points outward (away from water), making the meniscus concave (curved upward). Step 3: A concave meniscus causes capillary RISE in water. Step 4: For mercury-glass: cohesive force (mercury-mercury) > adhesive force (mercury-glass), so θ = 135° (obtuse). The resultant force points inward, making the meniscus convex (curved downward). Step 5: A convex meniscus causes capillary DEPRESS