Explain why
(a) To keep a piece of paper horizontal, you should blow over, not under, it
(b) When we try to close a water tap with our fingers, fast jets of water gush through the openings between our fingers
(c) The size of the needle of a syringe controls flow rate better than the thumb pressure exerted by a doctor while administering an injection
(d) A fluid flowing out of a small hole in a vessel results in a backward thrust on the vessel
(e) A spinning cricket ball in air does not follow a parabolic trajectory.

Question

The volume contraction of a solid copper cube of edge length 10 cm, when subjected to a hydraulic pressure of $ 7 \times 10^6 $ Pa, would be ________________________ mm$^3$. (Given bulk modulus of copper = $ 1.4 \times 10^{11} $ N m$^{-2}$)

Answer 1

Bernoulli's Equation: $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$

Fast flow → Low pressure. Slow flow → High pressure.

(a) Paper Lifts When Blowing Over It

Why over, not under?

Blowing under: High pressure pushes paper down
Blowing over: Fast air → low pressure above paper
Atmospheric pressure below > low pressure above → paper lifts up

$$P_\text{top (fast)} < P_\text{bottom (still)} \quad \Rightarrow \quad \text{lift}$$

(b) Water Jets Through Finger Gaps

Why fast jets escape?

Fingers block most flow → water squeezes through small gaps
Small area → high velocity (continuity: $A_1v_1 = A_2v_2$)
High velocity → low pressure → strong jet emerges

$$A_\text{gap} \ll A_\text{tap} \quad \Rightarrow \quad v_\text{gap} \gg v_\text{tap}$$

(c) Needle Size > Thumb Pressure

Why needle controls flow?

Poiseuille's law: Flow $\propto \frac{r^4}{\eta L}$ (radius to 4th power!)
Smaller needle → dramatically less flow regardless of thumb pressure
Thumb pressure overcome by viscous resistance in tiny needle

$$Q \propto r^4 \quad \Rightarrow \quad 0.5r \text{ gives } \frac{1}{16} \text{ flow}$$

(d) Vessel Thrust from Hole

Why backward push? Momentum conservation

Water exits right → gains rightward momentum
Vessel gains equal leftward momentum → recoils backward
Like rocket thrust: mass flow creates reaction force

$$F_\text{thrust} = \dot{m} v_\text{exit} \quad \text{(backward on vessel)}$$

(e) Cricket Ball Curve (Magnus Effect)

Why non-parabolic path?

Spin creates velocity difference above/below ball
Top: ball spin + air speed → fast flow → low pressure
Bottom: ball spin opposes air → slow flow → high pressure
Pressure difference → sideways lift force

$$F_\text{Magnus} \propto \omega \times v \quad \text{(curved trajectory)}$$

Core Principle Summary

Phenomenon	Key Bernoulli Effect
(a) Paper	Fast air over → suction
(b) Water jet	Constriction → acceleration
(c) Syringe	Small $r^4$ → low flow
(d) Vessel	Momentum recoil
(e) Cricket	Spin-induced pressure difference

Solution and Explanation