Mechanical Properties of Fluids — JEE Mains Physics

Formulas (3)

• Pressure at a depth, absolute and gauge · Pressure at depth h
  $$P = P_0 + \rho g h$$
• Pascal's law and the hydraulic lift · Hydraulic lift
  $$\frac{F_1}{A_1} = \frac{F_2}{A_2}, \qquad F_1 d_1 = F_2 d_2$$
• Floating bodies and Archimedes' principle · Submerged fraction
  $$\frac{V_{\text{sub}}}{V} = \frac{\rho_b}{\rho_l}$$

Watch out for (7)

• Doubling the depth does not double the pressure
  → Pressure at a depth, absolute and gauge
• Keep P₀ in a force on a base when the stem gives it
  → Pressure at a depth, absolute and gauge
• Across a partition, the atmosphere cancels
  → Pressure at a depth, absolute and gauge
• The small piston carries the same pressure
  → Pascal's law and the hydraulic lift
• A lift multiplies force, not work
  → Pascal's law and the hydraulic lift
• Body over liquid, not liquid over body
  → Floating bodies and Archimedes' principle
• A load adds displaced liquid, not displaced wood
  → Floating bodies and Archimedes' principle

Formulas (3)

• Continuity and Bernoulli's equation in a pipe · Continuity and Bernoulli
  $$A_1 v_1 = A_2 v_2, \qquad P + \rho g h + \tfrac{1}{2}\rho v^{2} = \text{constant}$$
• Speed from a pressure drop: gauges and wings · Lift on a wing
  $$F = \tfrac{1}{2}\rho\left(v_{\text{top}}^{2} - v_{\text{bottom}}^{2}\right)A$$
• Liquid leaving a hole: Torricelli's law · Torricelli's law and the range
  $$v = \sqrt{2gh}, \qquad x = 2\sqrt{h\,y}$$

Watch out for (7)

• Put both areas in the same unit
  → Continuity and Bernoulli's equation in a pipe
• The narrow section is at the lower pressure
  → Continuity and Bernoulli's equation in a pipe
• Square the speeds, then subtract
  → Continuity and Bernoulli's equation in a pipe
• Two wings: add both areas
  → Speed from a pressure drop: gauges and wings
• Convert km/h before squaring
  → Speed from a pressure drop: gauges and wings
• Depth is measured from the free surface
  → Liquid leaving a hole: Torricelli's law
• A load adds W/A, the atmosphere adds nothing
  → Liquid leaving a hole: Torricelli's law

Formulas (2)

• Newton's law of viscosity and the Reynolds number · Viscous force and Reynolds number
  $$F = \eta A\frac{v}{d}, \qquad R_e = \frac{\rho v d}{\eta}$$
• Stokes' law and the viscous force at constant velocity · Viscous force at constant velocity
  $$F_v = mg\left(1 - \frac{\rho_0}{\rho}\right) = \tfrac{4}{3}\pi r^{3}(\rho - \rho_0)g$$

Watch out for (4)

• Liquids and gases go opposite ways with temperature
  → Newton's law of viscosity and the Reynolds number
• Convert km/h and poise first
  → Newton's law of viscosity and the Reynolds number
• Buoyancy is subtracted, not added
  → Stokes' law and the viscous force at constant velocity
• The ball's density goes in the denominator
  → Stokes' law and the viscous force at constant velocity

Formulas (2)

• The terminal-velocity formula · Terminal velocity
  $$v_T = \frac{2r^{2}(\rho - \sigma)g}{9\eta}$$
• How terminal velocity scales with the radius · Merging drops
  $$v_T \propto r^{2}, \qquad R = n^{1/3}r \Rightarrow v' = n^{2/3}v$$

Watch out for (6)

• Diameter or radius
  → The terminal-velocity formula
• Do not mix CGS and SI
  → The terminal-velocity formula
• A faster launch does not change η
  → The terminal-velocity formula
• Same mass is not same material
  → How terminal velocity scales with the radius
• Merging multiplies by n^(2/3), not n
  → How terminal velocity scales with the radius
• v grows as r², not inversely with r
  → How terminal velocity scales with the radius

Formulas (2)

• Drops that merge or split · Splitting a drop into n
  $$W = 4\pi R^{2}T\left(n^{1/3} - 1\right)$$
• Work to blow a soap bubble · Soap bubble
  $$W = 8\pi T\left(r_2^{2} - r_1^{2}\right)$$

Watch out for (5)

• A diameter in the stem
  → Drops that merge or split
• The ratio of energies is n^(1/3), not n
  → Drops that merge or split
• Comparing two splittings uses n^(1/3) − 1
  → Drops that merge or split
• Count both faces of a soap film
  → Work to blow a soap bubble
• Diameters in the stem
  → Work to blow a soap bubble

Formulas (2)

• Excess pressure inside drops and bubbles · Excess pressure
  $$\Delta P = \frac{2T}{r}\ \text{(one surface)}, \qquad \Delta P = \frac{4T}{r}\ \text{(soap bubble)}$$
• Capillary rise · Capillary rise
  $$h = \frac{2T\cos\theta}{\rho g r}$$

Reference tables (1)

Watch out for (8)

• An air bubble in a liquid has ONE surface
  → Excess pressure inside drops and bubbles
• Add the depth term
  → Excess pressure inside drops and bubbles
• The common surface uses the difference of the radii
  → Excess pressure inside drops and bubbles
• Bores are often given as diameters
  → Capillary rise
• A tilted tube keeps the vertical height
  → Capillary rise
• The contact angle belongs to the pair
  → Contact angle, meniscus shape and surface-tension facts
• Soap water has LOWER surface tension
  → Contact angle, meniscus shape and surface-tension facts
• Gases are less viscous than liquids
  → Contact angle, meniscus shape and surface-tension facts