Q.Explain why

1. Concept: Bernoulli's Principle — where speed is high, pressure is low. Blowing over the paper creates fast-moving air above it, lowering the pressure there. The still air below exerts a higher pressure, pushing the paper upward and keeping it horizontal.
2. Concept: Bernoulli's Principle — the water speeds up as it passes through the narrow gaps between your fingers. This increased speed lowers the pressure locally, so the water is forced out in fast jets rather than spreading evenly.
3. Concept: Equation of Continuity — for a given volume flow rate, the speed of the fluid is inversely proportional to the cross-sectional area. The needle's tiny opening forces a high exit speed, which controls the jet's penetration and spread far more than the thumb pressure alone can.
4. Concept: Newton's Third Law / Conservation of Momentum — as fluid is ejected forward through the hole, an equal and opposite momentum is imparted to the vessel, pushing it backward (reaction thrust). …

Let’s take each case one by one, building the physics from the ground up.

(a) To keep a piece of paper horizontal, you should blow over, not under, it.

Concept: Bernoulli’s principle says that in a moving fluid, faster flow means lower pressure. When you blow air over the top of a paper strip, the air speed above is much higher than the still air below. This creates a pressure difference: lower pressure above, higher pressure below. The net upward force lifts the paper.

If you blew under the paper, the fast-moving air below would create low pressure there, and the still air above would push the paper down — making it sag or fall.

Step-by-step:

1. Identify the two regions: Above the paper (where you blow) and below (still air).
2. Apply Bernoulli: $P_{\text{above}} + \frac{1}{2}\rho v_{\text{above}}^2 = P_{\text{below}} + \frac{1}{2}\rho v_{\text{below}}^2$. Since $v_{\text{above}} > v_{\text{below}} \approx 0$, we get $P_{\text{above}} < P_{\text{below}}$.
3. Net force: Pressure difference $\Delta P = P_{\text{below}} - P_{\text{above}}$ acts upward over the paper’s area, lifting it.

(b) When we try to close a water tap with our fingers, fast jets of water gush through the openings between our fingers.

Concept: As you press your fingers against the tap opening, you reduce the area available for water to flow. For a given flow rate (volume per second), a smaller cross-sectional area forces the water speed to increase dramatically — by the continuity equation $A_1 v_1 = A_2 v_2$. The high-speed water then has low pressure (Bernoulli), so it doesn’t spread out; instead, it forms narrow, fast jets that shoot through the gaps.

Step-by-step:

1. Continuity equation: The tap supplies water at a roughly constant volume flow rate $Q$. If you block most of the opening, the remaining gaps have total area $A_{\text{gap}} \ll A_{\text{tap}}$. So $v_{\text{gap}} = Q / A_{\text{gap}}$ becomes very large.
2. Bernoulli’s effect: High speed in the gaps means low pressure there. The surrounding atmospheric pressure is higher, so the jet stays narrow and doesn’t spread — it shoots out as a fast stream.
3. Why it feels forceful: The momentum of the fast water (mass × high velocity) delivers a strong impulse per second, making the jets feel like they’re “gushing.”

(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.

Concept: Flow through a narrow tube (like a needle) is governed by Poiseuille’s law for viscous fluids: $Q = \frac{\pi r^4 \Delta P}{8 \eta L}$. The flow rate $Q$ depends on the fourth power of the needle’s radius $r$. Halving the radius reduces flow by a factor of 16 — a huge change. In contrast, thumb pressure $\Delta P$ appears only linearly, so doubling the pressure only doubles the flow. The needle’s size is therefore the dominant control.

Step-by-step:

1. Poiseuille’s law: For laminar flow of a viscous fluid through a cylindrical tube,

where $r$ = radius, $\Delta P$ = pressure difference, $\eta$ = viscosity, $L$ = length.

2. Compare sensitivities:

• Changing $r$: $Q \propto r^4$ — a tiny change in radius has a massive effect.
• Changing $\Delta P$: $Q \propto \Delta P$ — linear, much weaker.

3. Practical implication: A doctor’s thumb can vary pressure only within a limited range (say 2× to 3×), but switching from a 22-gauge needle to a 26-gauge needle (smaller radius) can reduce flow by over 90%. So needle size is the real flow controller.

(d) A fluid flowing out of a small hole in a vessel results in a backward thrust on the vessel.

Concept: This is Newton’s third law in action — the same principle that makes a rocket or a fire hose recoil. As fluid is ejected backward (relative to the vessel), the vessel experiences an equal and opposite forward force. But here, the fluid exits through a small hole, so the jet’s momentum change creates a reaction force on the vessel.

Step-by-step: …

1. Bernoulli: faster air above paper -> lower pressure above -> paper pushed up.
2. continuity: smaller gap area, same flow rate -> higher exit speed, narrow jets.
3. Poiseuille's law Q proportional to r^4 -- needle radius dominates over thumb-controlled pressure. …