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MHT-CET Physics · Formula sheet

Mechanical Properties of Fluids formulas

12 formulas, 1 reference table and 14 common traps for MHT-CET Physics Mechanical Properties of Fluids, grouped by subtopic.

Full notes with worked examples

Pressure with Depth and Buoyancy

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Pressure with Depth, and the Rising Bubble

Rising bubble

(P0+ρgd) V1=P0 V2  ⇒  ρgd=P0(V2V1−1)(P_0 + \rho g d)\,V_1 = P_0\,V_2 \;\Rightarrow\; \rho g d = P_0\left(\frac{V_2}{V_1} - 1\right)

Upthrust, and How Deep a Light Body Sinks

Maximum depth

dmax⁡=2gh2 (σ−ρ)g/ρ=hρσ−ρd_{\max} = \frac{2gh}{2\,(\sigma - \rho)g/\rho} = \frac{h\rho}{\sigma - \rho}

Common traps

Using 8H instead of 7H

The bottom pressure is 8 atmospheres, but one of them is the atmosphere itself. The water column is the other seven: depth 7H.

Writing the deceleration as (σ − ρ)g

The net force (σ − ρ)Vg acts on the body's own mass ρV, so the deceleration is (σ − ρ)g/ρ. Dropping the ρ loses the h ρ in the answer.

Surface Tension, Surface Energy, and Drops that Split or Merge

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Surface Tension as a Force Along a Line

Film between two plates

F=2TAt=2TA2VF = \frac{2TA}{t} = \frac{2TA^2}{V}

Work on Films and Soap Bubbles

Soap bubble

W=2×4πR2 T=8πR2TW = 2 \times 4\pi R^2\,T = 8\pi R^2 T

Drops that Split or Merge

Splitting a drop

W=4πR2T(n1/3−1),EdropletsEdrop=n1/3W = 4\pi R^2 T\left(n^{1/3} - 1\right), \qquad \frac{E_{\text{droplets}}}{E_{\text{drop}}} = n^{1/3}

Surface Molecules, Temperature and Impurities

ChangeSurface tensionAngle of contact
Temperature risesdecreases—
Critical temperaturezero—
Soap / soluble impurity addeddecreasesdecreases
A highly soluble salt can raise T slightly; the paper's answer is 'decreases' for soap and detergents.
Surface energy per unit area = surface tension.

Common traps

Counting only the outer rim of a ring

A disc with a hole touches the liquid along two circles. Both pull: 2πT(R + r), not 2πT(R − r).

Forgetting the second surface

A soap film or bubble has two faces. A liquid DROP has one. Using 4πR²T for a bubble halves the answer.

Using n^(2/3) for the total surface

Each droplet's area scales as n^(−2/3), and there are n of them: n × n^(−2/3) = n^(1/3). The factor n^(2/3) is the surface of the MERGED drop measured in droplet units, which is where E(n − n^(2/3)) comes from.

Thinking surface molecules have LESS energy

They have fewer neighbours, so fewer attractive bonds holding them down — their potential energy is higher, not lower.

Excess Pressure in Drops and Bubbles, and Capillary Rise

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Excess Pressure Inside a Drop and a Bubble

Excess pressure

ΔPdrop=2Tr,ΔPbubble=4Tr\Delta P_{\text{drop}} = \frac{2T}{r}, \qquad \Delta P_{\text{bubble}} = \frac{4T}{r}

Capillary rise

h=2Tcos⁡θrρgh = \frac{2T\cos\theta}{r\rho g}

Common traps

Taking 1.01 : 1.02 as the pressure ratio

Radius depends on the EXCESS pressure. Subtract the outside 1 atm first: 0.01 : 0.02, not 101 : 102.

Using 2T/r for a soap bubble

A bubble has an inner and an outer surface, so 4T/r. A drop, or the meniscus in a tube, has one: 2T/r.

Thinking the narrow tube holds more water

The narrow tube lifts liquid HIGHER (h ∝ 1/r) but holds LESS of it (m ∝ r). Radius r/5 gives 5h but m/5.

Viscosity, Stokes' Law and Terminal Velocity

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Viscous Force Between Layers, Reynolds Number and Critical Velocity

Reynolds number and critical velocity

Re=ρvdη,vc=Re ηρdR_e = \frac{\rho v d}{\eta}, \qquad v_c = \frac{R_e\,\eta}{\rho d}

Terminal Velocity

Terminal velocity (Stokes)

vt=2r2(ρ−σ)g9ηv_t = \frac{2r^2(\rho - \sigma)g}{9\eta}

Common traps

Thinking a viscous liquid turns turbulent sooner

Viscosity damps disturbances. The critical velocity is proportional to η: honey stays streamline at speeds where water would be turbulent.

Scaling terminal velocity with volume

v_t ∝ r², not r³. Two drops merging double the VOLUME, raise the radius by 2^(1/3), and raise the speed by 2^(2/3) ≈ 1.59, not 2.

Streamline Flow: Continuity, Bernoulli and Torricelli

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Continuity: Narrow Pipe, Faster Flow

Continuity

A1v1=A2v2=QA_1 v_1 = A_2 v_2 = Q

Bernoulli's Theorem

Bernoulli, horizontal pipe

P1+12ρv12=P2+12ρv22P_1 + \tfrac{1}{2}\rho v_1^2 = P_2 + \tfrac{1}{2}\rho v_2^2

Torricelli: Jets and Draining Tanks

Torricelli

v=2gh,tdrain∝hv = \sqrt{2gh}, \qquad t_{\text{drain}} \propto \sqrt{h}

Common traps

Scaling speed with radius, not area

Q = Av and A ∝ r². A third of the radius means a ninth of the area and nine times the speed, not three.

Putting the high pressure at the narrow part

Squeezing the pipe speeds the fluid up, and faster flow means LOWER pressure. The narrowest section has maximum speed and minimum pressure.

Measuring depth from the bottom

h in √(2gh) is the depth of the hole BELOW THE FREE SURFACE, not its height above the base. The hole nearest the bottom gives the fastest jet.

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