MECHANICAL PROPERTIES OF FLUIDS MCQs for NEET UG PHYSICS — Medium, Page 7

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Analyzing +4 | -1

Given below are two statements: One is labelled as Assertion (A) and other is labelled as Reason (R).
Assertion (A) : Time period of oscillation of a liquid drop depends on surface tension

(S)

, if density of the liquid is

\rho

and radius of the drop is

r

, then

T=K \sqrt{\frac{\rho r^3}{S^{3 / 2}}}

is dimensionally correct, where

K

is dimensionless.
Reason (R) : Using dimensional analysis we get R.H.S. having different dimension than that of time period.
In the light of above statements, choose the correct answer from the options given below.

View answer and explanation

Correct option: D

(A) is false but (R) is true

Option D is correct. Assertion (A) is dimensionally incorrect, and Reason (R) correctly states that dimensional analysis shows the incorrectness.

Solution approach

Correct Answer + Core Reason: (A) is false but (R) is true. Dimensional analysis shows the given formula for the time period is incorrect.

Step-by-Step Solution:

Step 1: Determine the dimensions of each quantity:
- Time period (T): [T]
- Surface tension (S): [MT $^{-2}$ ]
- Density ( $\rho$ ): [ML $^{-3}$ ]
- Radius (r): [L]

Step 2: Analyze the dimensions of the given formula: $T = K \sqrt{\frac{\rho r^3}{S^{3/2}}}$

Step 3: Substitute the dimensions:
[T] = $\sqrt{\frac{[ML^{-3}][L^3]}{[MT^{-2}]^{3/2}}} = \sqrt{\frac{[M]}{[M^{3/2}T^{-3}]}} = \sqrt{[M^{-1/2}T^3]} = [M^{-1/4}T^{3/2}]$

Step 4: Compare the dimensions: The dimensions on the left-hand side (LHS) are [T], while the dimensions on the right-hand side (RHS) are [M $^{-1/4}$ T $^{3/2}$ ]. Since the dimensions are not the same, the formula is dimensionally incorrect.

Step 5: Conclusion: Assertion (A) is false because the formula is dimensionally incorrect. Reason (R) is true because dimensional analysis confirms the incorrectness.

Why other options are wrong:
- Option A: Assertion (A) is false, so this option is incorrect.
- Option B: Assertion (A) is false, so this option is incorrect.
- Option C: Assertion (A) is false, but Reason (R) is true, so this option is incorrect.

Key Concepts:
- Dimensional analysis
- Dimensions of physical quantities (Time period, Surface Tension, Density, Radius)

Memory Aids:
- Remember the dimensions of basic quantities: Mass [M], Length [L], Time [T]

Quick Tips:
- Always check the dimensional correctness of a formula before using it.

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Understanding +4 | -1

The weight of an aeroplane flying in the air is balanced by

View answer and explanation

Correct option: B

Force due to reaction of gases ejected by the revolving propeller

Option A (Vertical component of the thrust created by air currents striking the lower surface of the wings) is incorrect. Option D (Force due to the pressure difference between the upper and lower surfaces of the wings created by different air speeds on the surfaces) is the primary way an airplane stays aloft.

Solution approach

Correct Answer + Core Reason: Force due to reaction of gases ejected by the revolving propeller. This is incorrect. The lift is primarily due to the pressure difference on the wings.

Step-by-Step Solution:

Step 1: Understanding Lift: Lift is the force that opposes the weight of an airplane, allowing it to stay in the air.

Step 2: Key Concept: Bernoulli's Principle and Airfoil Design. Airplane wings (airfoils) are designed to create a pressure difference between the upper and lower surfaces. The air travels faster over the curved upper surface, resulting in lower pressure, while the air travels slower under the flatter lower surface, resulting in higher pressure.

Step 3: Pressure Difference: This pressure difference creates an upward force (lift) that counteracts the weight of the airplane.

Step 4: Thrust: Thrust is the force that propels the airplane forward, usually generated by engines and propellers or jet engines. While thrust is necessary for flight, it's the pressure difference on the wings that primarily balances the weight.

Why other options are wrong:
- Option A (Vertical component of the thrust created by air currents striking the lower surface of the wings): While there is some upward force from the air striking the lower surface, it's a minor contribution compared to the pressure difference.
- Option B (Force due to reaction of gases ejected by the revolving propeller): This describes thrust, which propels the plane forward, but doesn't directly balance the weight.
- Option C (Upthrust of the air which will be equal to the weight of the air having the same volume as the plane): This is Archimedes' principle, which applies to buoyancy in fluids. While air is a fluid, the upthrust is negligible compared to the lift generated by the wings.

Key Concepts:
- Lift
- Bernoulli's Principle
- Airfoil Design
- Pressure Difference

Memory Aids:
- Faster air = Lower pressure = Lift

Quick Tips:
- Remember that the shape of the wings is crucial for generating lift.
- Understand the relationship between air speed and pressure.

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Application +4 | -1

Two cylindrical vessels with their bases at the same level, contain a liquid of density

\rho

. The height of the liquid in the vessel is

{h_1}

and in the other is

{h_2}

. The area of either base is A. The work done by the gravity in equalizing the levels when the vessels are connected.

View answer and explanation

Correct option: D

\frac{{A\rho g}}{4}{\left[ {{h_1} - {h_2}} \right]^2}

Option D (

\frac{{A\rho g}}{4}{\left[ {{h_1} - {h_2}} \right]^2}

) is correct because the work done by gravity is equal to the change in potential energy of the liquid as it equalizes its level.

Solution approach

Correct Answer + Core Reason: $\frac{{A\rho g}}{4}{\left[ {{{{h_1} - {h_2}}}} \right]^2}$ . The work done by gravity is equal to the change in potential energy of the liquid.

Step-by-Step Solution:

Step 1: When the vessels are connected, the liquid levels equalize. The final height in each vessel is $h = \frac{h_1 + h_2}{2}$ .

Step 2: The volume of liquid that moves from the first vessel to the second is $V = A(h_1 - h) = A(h_1 - \frac{h_1 + h_2}{2}) = A(\frac{h_1 - h_2}{2})$ .

Step 3: The height by which the center of mass of the liquid in the first vessel falls is $\frac{h_1 - h_2}{2}$ .

Step 4: The potential energy lost by the liquid in the first vessel is $PE_1 = \rho V g \frac{h_1 - h_2}{2} = \rho A(\frac{h_1 - h_2}{2}) g \frac{h_1 - h_2}{2} = \frac{A \rho g (h_1 - h_2)^2}{4}$ .

Step 5: The work done by gravity is equal to the potential energy lost: $W = \frac{A \rho g (h_1 - h_2)^2}{4}$ .

Step 6: Final Answer: $\frac{{A\rho g}}{4}{\left[ {{h_1} - {h_2}} \right]^2}$

Why other options are wrong:
- **Option A ( $\frac{{A\rho g{{\left( {{h_1} - {h_2}} \right)}^2}}}{2}$ ): This is incorrect. It misses a factor of 1/2 in the calculation.
- Option B ( $\frac{{A\rho g}}{2}\left[ {h_1^2 - h_2^2} \right]$ ): This is incorrect. It uses the difference of squares instead of the square of the difference.
- Option C ( $\frac{{A\rho g}}{4}\left[ {h_1^2 - h_2^2} \right]$ ): This is incorrect. It uses the difference of squares instead of the square of the difference.

Key Concepts:
- Potential Energy
- Work Done by Gravity

Memory Aids:
- Remember that the work done by gravity is equal to the change in potential energy.

Quick Tips:**
- Calculate the change in potential energy by considering the change in height of the center of mass.

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Analyzing +4 | -1

An ice block floats in a liquid whose density is less than water. A part of block is outside the liquid. When whole of ice has melted, the liquid level will

View answer and explanation

Correct option: B

Go down

Option B (Go down) is correct because the density of ice is less than water, and the liquid's density is even less than that of water. Upon melting, the water formed occupies less volume than the ice block displaced in the liquid.

Solution approach

Correct Answer + Core Reason: Go down. The liquid level decreases because the water formed from the melted ice occupies less volume than the ice displaced in the liquid, which itself was less dense than water.

Step-by-Step Solution:

Step 1: Initial State: Ice block floating in a liquid less dense than water, with part of the ice outside the liquid.

Step 2: Key Concept: Archimedes' Principle and Density. The ice displaces a volume of liquid whose weight equals the weight of the ice. When the ice melts, it turns into water.

Step 3: Density Comparison: Density of liquid < Density of water. Since the liquid is less dense than water, the volume of liquid displaced by the ice is greater than the volume of water the ice will become upon melting.

Step 4: Volume Change: When ice melts, its volume decreases because water is denser than ice. Since the liquid is even less dense than water, the volume of liquid displaced by the ice is greater than the volume of water formed after melting.

Step 5: Liquid Level: Because the volume of water formed is less than the volume of liquid initially displaced, the liquid level will go down.

Why other options are wrong:
- Option A (Rise): This would be true if the liquid were denser than water, but it's given that the liquid is less dense than water.
- Option C (Remain same): This would be true only if the density of the liquid was equal to the density of water, which is not the case here.
- Option D (First rise then go down): There's no mechanism for the liquid level to initially rise; it will only go down or remain the same.

Key Concepts:
- Archimedes' Principle
- Density and Volume relationship
- Buoyancy

Memory Aids:
- Density = Mass/Volume
- Less dense objects float higher.

Quick Tips:
- Consider the density relationships between the ice, water, and the liquid.
- Remember that melting ice results in a volume change.

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Understanding +4 | -1

For a ball falling in a liquid with constant velocity, ratio of the resistance force due to the liquid to that due to gravity is

View answer and explanation

Correct option: A

1

Option A (1) is correct because when an object falls with constant velocity, the net force acting on it is zero, meaning the resistance force is equal to the gravitational force.

Solution approach

Correct Answer + Core Reason: 1. When the ball falls with constant velocity, the net force is zero, so resistance force equals gravity.

Step-by-Step Solution:

Step 1: Understand the concept of terminal velocity.

Step 2: When a ball falls with constant velocity (terminal velocity), the net force acting on it is zero.

Step 3: The forces acting on the ball are: gravitational force (downwards) and resistance force due to the liquid (upwards).

Step 4: Since the net force is zero, the resistance force is equal in magnitude to the gravitational force.

Step 5: Therefore, the ratio of the resistance force to the gravitational force is 1.

Why other options are wrong:
- Option B: $\frac{{2{a^2}\rho \mathrm{g}}}{{9{\eta ^2}}}$ - This is incorrect as it doesn't represent the ratio of forces when the velocity is constant.
- Option C: $\frac{{2{a^2}\left( {\rho - \sigma } \right)\mathrm{g}}}{{9\\;\eta }}$ - This is incorrect as it doesn't represent the ratio of forces when the velocity is constant.
- Option D: None of these - This is incorrect because option A is correct.

Key Concepts:
- Terminal Velocity
- Net Force
- Gravitational Force
- Resistance Force

Memory Aids:
- Constant velocity means net force is zero.

Quick Tips:
- Remember that at terminal velocity, forces are balanced.

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Understanding +4 | -1

Bernoulli’s principle is not involved in the working/explanation of

View answer and explanation

Correct option: C

Blades of a kitchen mixer

Option C (Blades of a kitchen mixer) is correct because kitchen mixer blades primarily use mechanical force to mix ingredients, not pressure differences or fluid dynamics principles like Bernoulli's.

Solution approach

Correct Answer + Core Reason: Blades of a kitchen mixer. Bernoulli's principle is not the primary mechanism for their operation.

Step-by-Step Solution:

Step 1: Bernoulli's Principle: States that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy.

Step 2: Movement of Spinning Ball: A spinning ball experiences a pressure difference due to the Magnus effect, which is related to Bernoulli's principle. The spinning causes air to move faster on one side and slower on the other, creating a pressure difference that causes the ball to curve.

Step 3: Carburetor of Automobile: A carburetor uses Bernoulli's principle to create a pressure drop that draws fuel into the air stream. The venturi in the carburetor narrows the airflow, increasing its speed and decreasing its pressure, which sucks fuel into the engine.

Step 4: Heart Attack: Bernoulli's principle can explain the reduction of blood flow in arteries narrowed by plaque. The increased velocity of blood through the narrowed section results in a decreased pressure, potentially leading to collapse of the artery.

Step 5: Blades of a Kitchen Mixer: Kitchen mixer blades primarily use mechanical force to mix ingredients. While fluid dynamics are involved, the primary mechanism is not based on pressure differences created by varying fluid speeds as described by Bernoulli's principle.

Why other options are wrong:
- Option A (Movement of spinning ball): The Magnus effect, which relies on Bernoulli's principle, explains the curved trajectory of a spinning ball.
- Option B (Carburetor of automobile): Carburetors use Bernoulli's principle to draw fuel into the air stream.
- Option D (Heart attack): Bernoulli's principle can explain reduced blood flow in narrowed arteries.

Key Concepts:
- Bernoulli's Principle
- Fluid Dynamics
- Pressure and Velocity relationship

Memory Aids:
- Bernoulli's principle: Faster air = lower pressure

Quick Tips:
- Consider the primary mechanism of each device or phenomenon.
- Identify whether pressure differences due to fluid speed are the driving force.

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Understanding +4 | -1

A drop of water breaks into two droplets of equal size. In this process, which of the following statements is correct?

View answer and explanation

Correct option: B

The sum of the masses of the two droplets is equal to mass of drop

Option B (The sum of the masses of the two droplets is equal to mass of drop) is correct because mass is conserved in a physical process like breaking a drop into smaller droplets.

Solution approach

Correct Answer + Core Reason: The sum of the masses of the two droplets is equal to mass of drop. Mass is conserved during the process.

Step-by-Step Solution:

Step 1: Initial State: A single drop of water.

Step 2: Final State: Two droplets of equal size formed from the original drop.

Step 3: Key Concept: Conservation of Mass. In a closed system, mass is neither created nor destroyed. Therefore, the total mass remains constant.

Step 4: Application: The mass of the original drop must equal the sum of the masses of the two droplets.

Why other options are wrong:
- Option A (The sum of the temperature of the two droplets together is equal to temperature of the original drop): Temperature is not necessarily conserved. The temperature of the droplets could change due to surface tension effects or heat transfer with the environment.
- Option C (The sum of the radii of the two droplets is equal to the radius of the drop): This is incorrect. Volume is conserved, not radius. The volume of the original drop equals the sum of the volumes of the two droplets. If $R$ is the radius of the original drop and $r$ is the radius of each droplet, then $\frac{4}{3}\pi R^3 = 2 \times \frac{4}{3}\pi r^3$ , so $R^3 = 2r^3$ , and $R = r \sqrt[3]{2}$ .
- Option D (The sum of the surface areas of the two droplets is equal to the surface area of the original drop): This is incorrect. The surface area increases when a single drop breaks into smaller droplets. This is because the total volume remains constant, but the surface-to-volume ratio increases.

Key Concepts:
- Conservation of Mass
- Surface Tension
- Volume and Surface Area relationships

Memory Aids:
- Mass is conserved in physical changes.

Quick Tips:
- Remember that mass is a conserved quantity.
- Consider the changes in surface area and volume.

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Analyzing +4 | -1

Assertion: When the stopcock over the

U

-tube having a large bubble at its one end and a small bubble at the other end is opened, air flows from the large bubble to the small bubble.
Reason: The pressure inside the large bubble is greater than that inside small bubble.

View answer and explanation

Correct option: D

If both assertion and reason are false

Option D (If both assertion and reason are false) is correct because the pressure inside a smaller bubble is actually greater than the pressure inside a larger bubble due to surface tension, and air flows from higher to lower pressure.

Solution approach

Correct Answer + Core Reason: If both assertion and reason are false. The pressure is higher in the smaller bubble, and air flows from small to large bubble.

Step-by-Step Solution:

Step 1: Analyze the assertion.\nThe assertion states that when the stopcock over the U-tube is opened, air flows from the large bubble to the small bubble. This is incorrect because the pressure inside the smaller bubble is higher than the pressure inside the larger bubble due to surface tension.

Step 2: Analyze the reason.\nThe reason states that the pressure inside the large bubble is greater than that inside the small bubble. This is incorrect because the pressure inside the smaller bubble is greater.

Step 3: Explain the pressure difference.\nThe excess pressure inside a bubble is given by $\Delta P = \frac{2T}{r}$ , where $T$ is the surface tension and $r$ is the radius of the bubble. Since the radius of the small bubble is smaller, the excess pressure inside the small bubble is greater.

Step 4: Determine the direction of air flow.\nAir flows from higher pressure to lower pressure. Therefore, air will flow from the small bubble to the large bubble.

Step 5: Conclusion.\nBoth the assertion and the reason are false.

Why other options are wrong:
- Option A (If both assertion and reason are true and reason is the correct explanation of the assertion): Both are false.
- Option B (If both assertion and reason are true but reason is not correct explanation of the assertion): Both are false.
- Option C (If assertion is true, but reason is false): Both are false.

Key Concepts:
- Surface tension
- Excess pressure inside a bubble

Memory Aids:
- Smaller bubbles have higher internal pressure.

Quick Tips:

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Remembering +4 | -1

What is the pressure head in Bernoulli's theorem? (

p=

pressure,

\rho=

density,

g=

acceleration due to gravity)

View answer and explanation

Correct option: C

\frac{p}{\rho g}

Option C (

\frac{p}{\rho g}

) is correct. The pressure head in Bernoulli's theorem represents the height of a liquid column that would exert the pressure p.

Solution approach

Correct Answer + Core Reason: $\frac{p}{\rho g}$ . The pressure head is the pressure divided by the product of density and gravity.

Step-by-Step Solution:

Step 1: Bernoulli's theorem states: $P + \frac{1}{2} \rho v^2 + \rho g h = constant$ .

Step 2: Each term in the equation has units of pressure (energy per unit volume).

Step 3: The term $P$ represents the pressure energy per unit volume.

Step 4: Dividing $P$ by $\rho g$ gives the pressure head: $\frac{P}{\rho g}$ . This represents the height of a column of fluid that would exert the pressure $P$ .

Why other options are wrong:
- Option A: $\frac{p}{\rho}$ is incorrect because it does not have the correct units for pressure head (length).
- Option B: $\frac{g p}{\rho}$ is incorrect because it does not have the correct units for pressure head (length).
- Option D: $p$ is incorrect because it represents the pressure itself, not the pressure head.

Key Concepts:
- Bernoulli's theorem
- Pressure head

Memory Aids:
- Pressure head = $\frac{P}{\rho g}$ .

Quick Tips:
- Pressure head has units of length.

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Application +4 | -1

The viscous force on a lead shot travelling down with velocity V in a liquid column is F. If the same lead shot travelling down with velocity 3V in an other vertical liquid column is opposed by viscous force 2F, the ratio of the coefficients of viscosities of the liquids is

View answer and explanation

Correct option: B

3\mathrm{ }:\mathrm{ }2

Option B (3 : 2) is correct because it is derived from comparing the viscous forces and velocities in the two scenarios using Stokes' Law.

Solution approach

Correct Answer + Core Reason: $3\mathrm{ }:\mathrm{ }2$ . The ratio of viscosities is found by comparing the viscous forces and velocities.

Step-by-Step Solution:

Step 1: Recall Stokes' Law for viscous force: $F = 6\pi \eta r v$ , where $\eta$ is the coefficient of viscosity, $r$ is the radius, and $v$ is the velocity.

Step 2: Let $\eta_1$ be the viscosity of the first liquid and $\eta_2$ be the viscosity of the second liquid. The radius $r$ is the same for both lead shots.

Step 3: Write the equations for the viscous forces in both cases:
$F_1 = 6\pi \eta_1 r V = F$
$F_2 = 6\pi \eta_2 r (3V) = 2F$

Step 4: Divide the second equation by the first equation:
$\frac{{6\pi \eta_2 r (3V)}}{{6\pi \eta_1 r V}} = \frac{{2F}}{F}$

Step 5: Simplify the equation:
$\frac{{3\eta_2}}{{\eta_1}} = 2$

Step 6: Solve for the ratio of viscosities:
$\frac{{\eta_1}}{{\eta_2}} = \frac{3}{2}$

Final Answer: $3:2$

Why other options are wrong:
- Option A (3:4): This ratio is incorrect based on the given information and Stokes' Law.
- Option C (4:5): This ratio is incorrect based on the given information and Stokes' Law.
- Option D (1:2): This ratio is incorrect based on the given information and Stokes' Law.

Key Concepts:
- Stokes' Law
- Viscosity

Memory Aids:
- Stokes' Law: $F = 6\pi \eta r v$

Quick Tips:
- Remember Stokes' Law for viscous force on a sphere.
- Pay attention to the given ratios and use them to set up equations.

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Application +4 | -1

A large open tank has two holes in the wall. One is a square hole of side L at a depth y from the top and the other is a circular hole of radius R at a depth of 9y from the top. When the tank is completely filled with water quantities of water flowing out per second from both the holes is the same . Then R is equal to

View answer and explanation

Correct option: C

\frac{L}{{\sqrt {3\pi } }}

Option C (

\frac{L}{{\sqrt {3\pi } }}

) is correct because equating the flow rates from both holes and solving for R gives this result.

Solution approach

Correct Answer + Core Reason: $\frac{L}{{\sqrt {3\pi } }}$ . The rate of flow from a hole is proportional to the area of the hole and the square root of the depth. Equating the flow rates from both holes and solving for R gives the correct answer.

Step-by-Step Solution:

Step 1: State Torricelli's Law: The speed of efflux from a hole at a depth $h$ is given by $v = \sqrt{2gh}$ , where $g$ is the acceleration due to gravity.

Step 2: Calculate the speed of efflux for the square hole: $v_1 = \sqrt{2gy}$ .

Step 3: Calculate the speed of efflux for the circular hole: $v_2 = \sqrt{2g(9y)} = \sqrt{18gy} = 3\sqrt{2gy}$ .

Step 4: Calculate the rate of flow for the square hole: $Q_1 = A_1 v_1 = L^2 \sqrt{2gy}$ .

Step 5: Calculate the rate of flow for the circular hole: $Q_2 = A_2 v_2 = \pi R^2 (3\sqrt{2gy})$ .

Step 6: Equate the rates of flow: $Q_1 = Q_2$ , so $L^2 \sqrt{2gy} = \pi R^2 (3\sqrt{2gy})$ .

Step 7: Simplify the equation: $L^2 = 3 \pi R^2$ .

Step 8: Solve for R: $R^2 = \frac{L^2}{3 \pi}$ , so $R = \sqrt{\frac{L^2}{3 \pi}} = \frac{L}{\sqrt{3 \pi}}$ .

Why other options are wrong:
- Option A: $3\pi L$ is incorrect because it does not follow from equating the flow rates.
- Option B: $\frac{L}{{3\pi }}$ is incorrect because it does not follow from equating the flow rates and solving for R.
- Option D: $\frac{L}{{\sqrt 3 \pi }}$ is incorrect because it contains a missing square root around the 3.

Key Concepts:
- Torricelli's Law
- Rate of flow
- Speed of efflux

Memory Aids:
- Rate of flow = Area x Velocity

Quick Tips:
- Remember Torricelli's Law for the speed of efflux.
- Equate the flow rates and solve for the unknown variable.

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Remembering +4 | -1

Surface tension vanishes at

View answer and explanation

Correct option: C

critical temperature

Option C is correct. Surface tension vanishes at the critical temperature because the distinction between liquid and gas phases disappears.

Solution approach

Correct Answer + Core Reason: critical temperature. Surface tension becomes zero at the critical temperature.

Step-by-Step Solution:

Step 1: Define Critical Temperature: The critical temperature (Tc) is the temperature above which a substance cannot exist in the liquid state, no matter how much pressure is applied. Above the critical temperature, the substance exists as a gas.

Step 2: Surface Tension and Phase Transition: Surface tension is a property that exists at the interface between a liquid and another phase (e.g., gas or another liquid). At the critical temperature, the distinction between the liquid and gas phases disappears, and the interface vanishes.

Step 3: Conclusion: Since surface tension exists at the interface between liquid and gas phases, it vanishes at the critical temperature when the interface disappears.

Why other options are wrong:
- Option A: Absolute zero temperature is the lowest possible temperature, but surface tension doesn't necessarily vanish there.
- Option B: Transition temperature refers to phase transitions in solids, not liquids.
- Option D: Surface tension does have a specific value at the critical temperature (zero).

Key Concepts:
- Surface tension
- Critical temperature
- Phase transition

Memory Aids:
- Critical temperature = No liquid-gas interface = No surface tension

Quick Tips:
- Remember that surface tension is related to the liquid-gas interface.

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Analyzing +4 | -1

A: If barometer is accelerated upwards, the level of mercury in the tube of barometer will decrease.
R: The effective value of '

{g}

' increases while the barometer is accelarated up.

View answer and explanation

Correct option: B

Both A and R are true and R is not the correct explanation of A

Option B (Both A and R are true and R is not the correct explanation of A) is correct. The assertion and reason are both true, but the reason does not explain the assertion. The decrease in mercury level is due to the pseudo force acting downwards.

Solution approach

Correct Answer + Core Reason: Both A and R are true and R is not the correct explanation of A. The mercury level decreases due to the pseudo force, and the effective g increases, but they are not directly related in this context.

Step-by-Step Solution:

Step 1: Analyze Assertion (A):\nWhen a barometer is accelerated upwards, a pseudo force acts on the mercury in the tube. This pseudo force acts in the downward direction, effectively increasing the weight of the mercury column. To balance this increased weight, the height of the mercury column must decrease.\nTherefore, the assertion is true.

Step 2: Analyze Reason (R):\nWhen a barometer is accelerated upwards with acceleration 'a', the effective acceleration due to gravity becomes g' = g + a. Thus, the effective value of 'g' increases.\nTherefore, the reason is true.

Step 3: Determine if R is the correct explanation of A:\nWhile both A and R are true, the reason R does not directly explain why the mercury level decreases. The decrease in mercury level is a consequence of the pseudo force acting on the mercury, not simply the increase in the effective value of 'g'. The increased 'g' is related to the pseudo force, but it's the pseudo force itself that directly causes the change in the mercury level.

Why other options are wrong:
- Option A: R is not the correct explanation of A.
- Option C: R is true.
- Option D: A is true.

Key Concepts:
- Pseudo force
- Effective acceleration due to gravity
- Barometer principle

Memory Aids:
- Remember to consider pseudo forces in non-inertial frames of reference.

Quick Tips:
- When analyzing situations in accelerating frames, always consider the pseudo forces.

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Application +4 | -1

A boat having length 2 m and width 1 m is floating in a lake. When a man stands on the boat, it is depressed by 3 cm. The mass of the man is

View answer and explanation

Correct option: C

60 kg

Option C (60 kg) is correct because the mass of the man equals the mass of the water displaced due to the boat's depression, calculated using Archimedes' principle.

Solution approach

Correct Answer + Core Reason: 60 kg. The mass of the man is equal to the mass of the water displaced by the boat.

Step-by-Step Solution:

Step 1: Identify the given information: length $l = 2 \text{{ m}}$ , width $w = 1 \text{{ m}}$ , depression $h = 3 \text{{ cm}} = 0.03 \text{{ m}}$ .

Step 2: Calculate the volume of water displaced: $V = lwh = (2 \text{{ m}})(1 \text{{ m}})(0.03 \text{{ m}}) = 0.06 \text{{ m}}^3$ .

Step 3: Calculate the mass of the water displaced: $m = \rho V$ , where $\rho = 1000 \text{{ kg/m}}^3$ is the density of water. So, $m = (1000 \text{{ kg/m}}^3)(0.06 \text{{ m}}^3) = 60 \text{{ kg}}$ .

Step 4: The mass of the man is equal to the mass of the water displaced: $m_{\text{{man}}} = 60 \text{{ kg}}$ .

Step 5: Final Answer: 60 kg.

Why other options are wrong:
- Option A (50 kg): This is an incorrect calculation of the displaced water mass, possibly due to using a wrong value for the depression or density.
- Option B (55 kg): This is also an incorrect calculation of the displaced water mass.
- Option D (70 kg): This is an overestimate of the mass, likely due to an error in volume or density calculation.

Key Concepts:
- Archimedes' principle: Buoyant force equals the weight of the fluid displaced.
- Density: $\rho = \frac{{m}}{{V}}$

Memory Aids:
- The weight of the displaced fluid equals the buoyant force.

Quick Tips:
- Ensure all units are consistent before calculations.

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Application +4 | -1

The pressure on a swimmer 20 m below the surface of water at sea level is

View answer and explanation

Correct option: D

3.0 atm

The pressure on a swimmer 20 m below the surface of water is 3.0 atm. This is calculated by adding the atmospheric pressure (1 atm) to the pressure due to the water column (

\rho g h

).

Solution approach

Correct Answer + Core Reason: 3.0 atm. The total pressure is the sum of atmospheric pressure and the pressure due to the water column.

Step-by-Step Solution:

Step 1: Calculate the pressure due to the water column: $P_{water} = \rho * g * h$ , where $\rho = 1000 kg/m^3$ , $g = 9.8 m/s^2 \approx 10 m/s^2$ , and $h = 20 m$ .

Step 2: $P_{water} = 1000 kg/m^3 * 10 m/s^2 * 20 m = 2 * 10^5 N/m^2 = 2 atm$ .

Step 3: Add the atmospheric pressure: $P_{total} = P_{atm} + P_{water} = 1 atm + 2 atm = 3 atm$ .

Why other options are wrong:
- Option A: 1.0 atm: Only considers atmospheric pressure, not the pressure due to the water column.
- Option B: 2.0 atm: Only considers the pressure due to the water column, not atmospheric pressure.
- Option C: 2.5 atm: Incorrect calculation.

Key Concepts:
- Fluid Pressure
- Atmospheric Pressure

Memory Aids:
- Remember to add atmospheric pressure to the pressure due to the fluid column.

Quick Tips:
- Always consider atmospheric pressure when calculating total pressure in a fluid.

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Understanding +4 | -1

Blood pressure rises in old age due to arteries becoming narrow. This phenomenon could be explained on the basis of

View answer and explanation

Correct option: C

Poiseuille's equation

Option C (Poiseuille's equation) is correct because it relates blood pressure to the radius of blood vessels, which narrows with age.

Solution approach

Correct Answer + Core Reason: Poiseuille's equation. Poiseuille's equation relates the flow rate of a fluid to the pressure difference, viscosity, and radius of the tube. Narrowing of arteries (decrease in radius) leads to an increase in blood pressure to maintain the same flow rate.

Step-by-Step Solution:

Step 1: State Poiseuille's Equation: $Q = \frac{\pi r^4 \Delta P}{8 \eta l}$ , where:
- $Q$ is the flow rate
- $r$ is the radius of the tube
- $\Delta P$ is the pressure difference
- $\eta$ is the viscosity
- $l$ is the length of the tube

Step 2: Rearrange the equation to solve for pressure difference: $\Delta P = \frac{8 \eta l Q}{\pi r^4}$ .

Step 3: Analyze the effect of narrowing arteries: As arteries become narrower in old age, the radius $r$ decreases. Since $\Delta P$ is inversely proportional to $r^4$ , a decrease in $r$ leads to a significant increase in $\Delta P$ , which is the blood pressure.

Why other options are wrong:
- Option A: Newton's laws are fundamental laws of motion and do not directly explain the relationship between blood pressure and artery radius.
- Option B: The photoelectric effect is a phenomenon related to the emission of electrons from a material when light shines on it and is not relevant to blood pressure.
- Option D: Van der Waal's equation describes the behavior of real gases and is not relevant to blood pressure.

Key Concepts:
- Poiseuille's equation
- Fluid dynamics
- Blood pressure
- Viscosity

Memory Aids:
- Remember that blood pressure is inversely proportional to the fourth power of the radius of the artery.

Quick Tips:
- Understand the relationship between flow rate, pressure, radius, and viscosity.
- Remember that small changes in radius can have a significant impact on pressure.

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Understanding +4 | -1

Two stretched membranes of area

2\mathrm{c}{\mathrm{m}^2}

and

3\mathrm{c}{\mathrm{m}^2}

are placed in a liquid at the same depth. The ratio of pressure on them is

View answer and explanation

Correct option: A

1\mathrm{ }:\mathrm{ }1

Option A (

1\mathrm{ }:\mathrm{ }1

) is correct because pressure in a liquid at a given depth depends only on the depth and the density of the liquid, not on the area of the surface.

Solution approach

Correct Answer + Core Reason: $1\mathrm{ }:\mathrm{ }1$ . Pressure at a given depth in a liquid is independent of the area.

Step-by-Step Solution:

Step 1: Recall the formula for pressure in a liquid: $P = \rho g h$ , where $\rho$ is the density of the liquid, $g$ is the acceleration due to gravity, and $h$ is the depth.

Step 2: The problem states that both membranes are placed at the same depth in the liquid. Therefore, $h$ is the same for both.

Step 3: Since $\rho$ and $g$ are also constant, the pressure $P$ is the same for both membranes.

Step 4: Therefore, the ratio of the pressure on them is $1:1$ .

Why other options are wrong:
- Option B: $2\mathrm{ }:\mathrm{ }3$ . This is incorrect because pressure is independent of area. This option incorrectly assumes pressure is proportional to area.
- Option C: $3\mathrm{ }:\mathrm{ }2$ . This is incorrect because pressure is independent of area. This option incorrectly assumes pressure is inversely proportional to area.
- Option D: ${2^2}:{3^2}$ . This is incorrect because pressure is independent of area. This option incorrectly squares the areas and assumes a relationship with pressure.

Key Concepts:
- Pressure in a Liquid
- Hydrostatic Pressure

Memory Aids:
- Pressure at a depth depends only on depth and density.

Quick Tips:
- Pressure is force per unit area, but at a given depth, the pressure itself is independent of the area.

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Remembering +4 | -1

The rate of flow of liquid in a tube of radius

r

, length

l

, whose ends are maintained at a pressure difference

p

is

V=\frac{\pi N p r^4}{\eta l}

, where

\eta

is coefficient of the viscosity and

N

is

View answer and explanation

Correct option: B

\frac{1}{8}

Option B (

\frac{1}{8}

) is correct because the correct formula for the rate of flow of liquid is

V=\frac{\pi p r^4}{8 \eta l}

.

Solution approach

Correct Answer + Core Reason: $\frac{1}{8}$ . The correct formula for the rate of flow of liquid in a tube is given by Poiseuille's equation, which has a factor of 8 in the denominator.

Step-by-Step Solution:

Step 1: Recall Poiseuille's equation: Poiseuille's equation describes the rate of flow of a viscous, incompressible fluid through a cylindrical pipe.

Step 2: Write the equation: $V = \frac{\pi p r^4}{8 \eta l}$ , where:
- $V$ is the rate of flow (volume per unit time)
- $p$ is the pressure difference between the ends of the tube
- $r$ is the radius of the tube
- $\eta$ is the coefficient of viscosity
- $l$ is the length of the tube

Step 3: Compare the given equation with Poiseuille's equation: $V=\frac{\pi N p r^4}{\eta l}$ .

Step 4: Identify the value of $N$ : By comparing the two equations, we can see that $N = \frac{1}{8}$ .

Why other options are wrong:
- Option A: $N = 8$ would give an incorrect equation.
- Option C: $N = 16$ would give an incorrect equation.
- Option D: $N = \frac{1}{16}$ would give an incorrect equation.

Key Concepts:
- Poiseuille's equation
- Viscosity
- Fluid dynamics

Memory Aids:
- Remember the "8" in the denominator of Poiseuille's equation.

Quick Tips:
- Know Poiseuille's equation by heart.
- Pay attention to the factors in the equation.

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Remembering +4 | -1

Increase in pressure at one point of the enclosed liquid in rest is transmitted equally to all other points of liquid". Above statement is in accordance with the

View answer and explanation

Correct option: B

Pascal's law

The correct answer is B (Pascal's law) because Pascal's law states that pressure applied to an enclosed fluid is transmitted equally to every point in the fluid.

Solution approach

Correct Answer + Core Reason: Pascal's law. Pascal's law describes the transmission of pressure in an enclosed liquid.

Step-by-Step Solution:

Step 1: Recall Pascal's Law: Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and to the walls of its container.

Step 2: Relate the given statement to Pascal's Law.

Step 3: Eliminate other options based on their definitions:
- Archimedes' principle deals with buoyancy.
- Equation of continuity deals with fluid flow.
- Bernoulli's theorem deals with the relationship between pressure, velocity, and height of a fluid.

Step 4: Conclusion: The statement is in accordance with Pascal's law.

Why other options are wrong:
- Option A: Archimedes' principle relates to buoyancy, not pressure transmission.
- Option C: Equation of continuity relates to fluid flow, not pressure transmission in a static fluid.
- Option D: Bernoulli's theorem relates to fluid dynamics, not pressure transmission in a static fluid.

Key Concepts:
- Pascal's Law
- Fluid Pressure
- Archimedes' Principle
- Equation of Continuity
- Bernoulli's Theorem

Memory Aids:
- Pascal's Law: Pressure is transmitted equally in all directions in a static fluid.

Quick Tips:
- Understand the fundamental principles of fluid mechanics.

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Analyzing +4 | -1

Check the dimensional consistency of the Poiseuille's formula for the laminar flow in a tube:

Q=\frac{\pi R^4\left(p_1-p_2\right)}{8 \eta l}

View answer and explanation

Correct option: A

{L}^3 {T}^{-1}

, Consistent

Option A (

{L}^3 {T}^{-1}

, Consistent) is correct because the dimensions on both sides of Poiseuille's formula match, indicating dimensional consistency.

Solution approach

Correct Answer + Core Reason: ${L}^3 {T}^{-1}$ , Consistent. Dimensional analysis confirms that both sides of Poiseuille's formula have the same dimensions.

Step-by-Step Solution:

Step 1: Write down Poiseuille's formula: $Q=\frac{\pi R^4(p_1-p_2)}{8 \eta l}$

Step 2: Determine the dimensions of each term:
- $Q$ (volume flow rate): Volume/Time = $L^3 T^{-1}$
- $R$ (radius): $L$
- $(p_1 - p_2)$ (pressure difference): Force/Area = $MLT^{-2} / L^2 = ML^{-1}T^{-2}$
- $\eta$ (viscosity): $ML^{-1}T^{-1}$
- $l$ (length): $L$

Step 3: Substitute the dimensions into the formula:
$Q = \frac{L^4 (ML^{-1}T^{-2})}{ML^{-1}T^{-1} L} = \frac{L^4 ML^{-1}T^{-2}}{MLT^{-1}} = L^3 T^{-1}$

Step 4: Compare the dimensions on both sides:
Left side: $Q = L^3 T^{-1}$
Right side: $L^3 T^{-1}$

Step 5: Since the dimensions on both sides are the same, the formula is dimensionally consistent.

Step 6: Final Answer: ${L}^3 {T}^{-1}$ , Consistent

Why other options are wrong:
- Option B: ${L}^4 {T}^{-2}$ , Inconsistent: The dimensions are incorrect, and the formula is consistent.
- Option C: ${L}^2 {T}^{5}$ , Consistent: The dimensions are incorrect, and the formula is consistent.
- Option D: ${L} {T}^{-1}$ , Inconsistent: The dimensions are incorrect, and the formula is consistent.

Key Concepts:
- Dimensional analysis
- Poiseuille's formula
- Dimensions of physical quantities (length, mass, time)

Memory Aids:
- Remember the dimensions of basic quantities like force, pressure, and viscosity.

Quick Tips:
- Always write down the dimensions of each term before substituting into the equation.
- Ensure that the dimensions on both sides of the equation are the same.

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Understanding +4 | -1

Assertion: Paper pins are made to have pointed end.
Reason: Because pointed pins have very small area due to which even for small applied force it exert large pressure on the surface.

View answer and explanation

Correct option: A

If both assertion and reason are true and reason is the correct explanation of the assertion

Option A (If both assertion and reason are true and reason is the correct explanation of the assertion) is correct. Pressure is inversely proportional to area, so a smaller area results in higher pressure for the same force.

Solution approach

Correct Answer + Core Reason: If both assertion and reason are true and reason is the correct explanation of the assertion. Pointed ends on paper pins concentrate the force over a very small area, resulting in a large pressure that allows the pin to easily penetrate the surface.

Step-by-Step Solution:

Step 1: Define pressure: Pressure is defined as force per unit area: $P = \frac{F}{A}$ , where $P$ is the pressure, $F$ is the force, and $A$ is the area.

Step 2: Analyze the relationship between pressure and area: From the equation, it is clear that for a given force $F$ , the pressure $P$ is inversely proportional to the area $A$ . This means that if the area decreases, the pressure increases, and vice versa.

Step 3: Apply this to the paper pin: A pointed end has a very small area. When a force is applied to the paper pin, this force is concentrated over the small area of the pointed end, resulting in a large pressure.

Step 4: Explain why this helps the pin penetrate the surface: The large pressure exerted by the pointed end is sufficient to overcome the resistance of the surface, allowing the pin to easily penetrate it.

Step 5: Therefore, both the assertion (paper pins are made to have pointed ends) and the reason (pointed pins have very small area due to which even for small applied force it exert large pressure on the surface) are true, and the reason is the correct explanation of the assertion.

Why other options are wrong:
- Option B: The reason is the correct explanation.
- Option C: The assertion is true.
- Option D: Both the assertion and reason are true.

Key Concepts:
- Pressure
- Force
- Area

Memory Aids:
- Pressure = Force/Area
- Smaller area, higher pressure (for the same force).

Quick Tips:
- Remember the relationship between pressure, force, and area.
- Consider how the area affects the pressure for a given force.

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Application +4 | -1

The total weight of a piece of wood is 6 kg. In the floating state in water its

\frac{1}{3}

part remains inside the water. On this floating solid, what maximum weight is to be put such that the whole of the piece of wood is to be drowned in the water?>

View answer and explanation

Correct option: A

12 kg

Option A (12 kg) is correct because the additional weight required to fully submerge the wood is equal to the weight of the water displaced by the remaining 2/3 of the wood's volume.

Solution approach

Correct Answer + Core Reason: 12 kg. The additional weight needed is the weight of water displaced by the remaining volume.

Step-by-Step Solution:

Step 1: Understand the floating condition.\nWhen the wood is floating, its weight is equal to the buoyant force. Let $V$ be the total volume of the wood. Since $\frac{1}{3}$ of the wood is inside the water, the volume submerged is $\frac{1}{3}V$ .\nThe buoyant force is equal to the weight of the water displaced, so $6 \text{ kg} = \frac{1}{3}V \rho_w g$ , where $\rho_w$ is the density of water (1000 kg/m $^3$ ) and $g$ is the acceleration due to gravity.

Step 2: Calculate the total volume of the wood.\nFrom the equation above, we have $V = \frac{3 \times 6}{\rho_w g} = \frac{18}{\rho_w g}$ .

Step 3: Determine the remaining volume to be submerged.\nThe remaining volume to be submerged is $V - \frac{1}{3}V = \frac{2}{3}V$ .

Step 4: Calculate the additional weight required.\nThe additional weight required is equal to the weight of the water displaced by the remaining volume: $W_{add} = \frac{2}{3}V \rho_w g = \frac{2}{3} \times \frac{18}{\rho_w g} \times \rho_w g = \frac{2}{3} \times 18 = 12 \text{ kg}$ .

Why other options are wrong:
- Option B (10 kg): Incorrect calculation of the additional weight.
- Option C (14 kg): Incorrect calculation of the additional weight.
- Option D (15 kg): Incorrect calculation of the additional weight.

Key Concepts:
- Buoyancy
- Archimedes' principle

Memory Aids:
- Buoyant force equals the weight of the displaced fluid.

Quick Tips:
- Remember that the weight of the floating object equals the buoyant force.

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Analyzing +4 | -1

A: An ice block with lead short is floating in water. When ice melt completely water level slightly drops.
R: When ice block with less shot is placed in water, volume of displaced water is more then that of the ice block

View answer and explanation

Correct option: B

Both A and R are true and R is not the correct explanation of A

Option B (Both A and R are true and R is not the correct explanation of A) is correct. The water level drops because the volume of water displaced by the melted ice is less than the volume of water displaced by the ice block with the lead shot. The reason is true but doesn't explain the drop in water level.

Solution approach

Correct Answer + Core Reason: Both A and R are true and R is not the correct explanation of A. The water level drops because the melted ice contributes less volume than the initial displacement, and the reason is a true statement but doesn't explain the drop.

Step-by-Step Solution:

Step 1: Analyze Assertion A: When the ice melts, it becomes water. The volume of this water is determined by the mass of the ice. The lead shot remains at the bottom.

Step 2: The initial volume of water displaced is determined by the combined weight of the ice and the lead shot.

Step 3: When the ice melts, the water formed has the same mass as the ice. However, the volume of water formed is less than the volume of water initially displaced by the ice because the density of water is greater than the effective density of the ice block (ice + lead).

Step 4: Therefore, the water level drops slightly.

Step 5: Analyze Reason R: When the ice block with lead shot is placed in water, the volume of displaced water is more than that of the ice block alone. This is because the lead shot increases the overall density, requiring more water to be displaced to achieve equilibrium.

Step 6: Reason R is true, but it does not explain why the water level drops when the ice melts. The water level drops because the volume of water produced by the melted ice is less than the volume of water initially displaced by the ice block and lead shot.

Why other options are wrong:
- Option A: Both A and R are true and R is the correct explanation of A is incorrect because R does not explain why the water level drops.
- Option C: A is true but R is false is incorrect because both A and R are true.
- Option D: Both A and R are false is incorrect because both A and R are true.

Key Concepts:
- Archimedes' principle
- Buoyancy
- Density

Memory Aids:
- The weight of the displaced fluid equals the weight of the object.

Quick Tips:
- Consider the density changes when ice melts into water.

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Application +4 | -1

The rate of flow of liquid through a capillary tube of radius

r

is

V

when the pressure difference across the two ends of the capillary is

p

. If pressure is increased by 3

p

and radius is reduced to

r/2

, then the rate of flow becomes

View answer and explanation

Correct option: C

V/4

Option C (

V/4

) is correct because the rate of flow is proportional to the pressure difference and the fourth power of the radius, so increasing the pressure by a factor of 4 and halving the radius results in a flow rate that is 1/4 of the original.

Solution approach

Correct Answer + Core Reason: $V/4$ . The rate of flow is proportional to pressure and the fourth power of the radius.

Step-by-Step Solution:

Step 1: State Poiseuille's equation.\nThe rate of flow of a liquid through a capillary tube is given by Poiseuille's equation:
$V = \frac{\pi p r^4}{8 \eta l}$ , where $V$ is the rate of flow, $p$ is the pressure difference, $r$ is the radius of the tube, $\eta$ is the viscosity of the liquid, and $l$ is the length of the tube.

Step 2: Identify the changes.\nThe pressure is increased by 3 $p$ , so the new pressure is $p' = p + 3p = 4p$ . The radius is reduced to $r/2$ , so the new radius is $r' = r/2$ .

Step 3: Calculate the new rate of flow.\nThe new rate of flow $V'$ is given by:
$V' = \frac{\pi p' (r')^4}{8 \eta l} = \frac{\pi (4p) (r/2)^4}{8 \eta l} = \frac{\pi (4p) (r^4/16)}{8 \eta l} = \frac{4}{16} \frac{\pi p r^4}{8 \eta l} = \frac{1}{4} V$

Step 4: State the final answer.\nThe new rate of flow is $V/4$ .

Why other options are wrong:
- **Option A ( $V/9$ ): This is incorrect because it doesn't correctly account for the effect of both the pressure increase and the radius decrease.
- Option B ( $3V/8$ ): This is incorrect because it doesn't correctly account for the effect of both the pressure increase and the radius decrease.
- Option D ( $V/3$ ): This is incorrect because it doesn't correctly account for the effect of both the pressure increase and the radius decrease.

Key Concepts:**
- Poiseuille's equation: $V = \frac{\pi p r^4}{8 \eta l}$

Memory Aids:
- Poiseuille's equation: Rate of flow depends on pressure and radius to the fourth power.

Quick Tips:
- Remember that the rate of flow is very sensitive to changes in the radius of the tube.

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Analyzing +4 | -1

Given below are two statements: one is labelled as
Assertion A and the other is labelled as Reason R

Assertion A : A spherical body of radius

\left( {5 \pm 0.1} \right)\,\,{\text{mm}}

having a particular density is falling through a
liquid of constant density. The percentage error in
the calculation of its terminal velocity is 4%.

Reason R : The terminal velocity of the spherical
body falling through the liquid is inversely
proportional to its radius.

In the light of the above statements, choose the
correct answer from the options given below

View answer and explanation

Correct option: C

A is true but R is false

Option C (A is true but R is false) is correct because the percentage error calculation is valid, but the terminal velocity is directly proportional to the square of the radius, not inversely proportional to the radius.

Solution approach

Correct Answer + Core Reason: A is true but R is false. The percentage error is correct, but the relationship between terminal velocity and radius is incorrect.

Step-by-Step Solution:

Step 1: Analyze Assertion A: The radius is given as $r = (5 \pm 0.1) \text{mm}$ . The percentage error in radius is $\frac{0.1}{5} \times 100 = 2\%$ .

Step 2: The terminal velocity is given by $v_t = \frac{2r^2(\rho - \sigma)g}{9\eta}$ , where $\rho$ is the density of the sphere and $\sigma$ is the density of the liquid.

Step 3: Since $v_t \propto r^2$ , the percentage error in $v_t$ is $2 \times$ the percentage error in $r$ . Therefore, the percentage error in $v_t$ is $2 \times 2\% = 4\%$ . Assertion A is true.

Step 4: Analyze Reason R: The terminal velocity $v_t$ is proportional to $r^2$ , not inversely proportional to $r$ . Therefore, Reason R is false.

Step 5: Conclusion: Assertion A is true, and Reason R is false.

Why other options are wrong:
- Option A (Both A and R are true and R is the correct explanation of A): This is incorrect because R is false.
- Option B (Both A and R are true but R is NOT the correct explanation of A): This is incorrect because R is false.
- Option D (A is false but R is true): This is incorrect because A is true and R is false.

Key Concepts:
- Terminal Velocity
- Error Analysis
- Proportionality

Memory Aids:
- Terminal velocity formula: $v_t = \frac{2r^2(\rho - \sigma)g}{9\eta}$

Quick Tips:
- Remember that percentage errors multiply when quantities are raised to a power.
- Carefully analyze the proportionality relationships in the given formulas.

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Remembering +4 | -1

Surface tension depends on

View answer and explanation

Correct option: D

The nature of the liquid

Option D (The nature of the liquid) is correct because surface tension is an intrinsic property of a liquid, dependent on intermolecular forces.

Solution approach

Correct Answer + Core Reason: The nature of the liquid. Surface tension is a property of the liquid itself.

Step-by-Step Solution:

Step 1: Recall the definition of surface tension.

Step 2: Surface tension is a property of liquids arising from intermolecular forces, specifically cohesion.

Step 3: It depends on the type of liquid and temperature, but not directly on the area or length considered.

Why other options are wrong:
- Option A: The area of the surface does not directly determine surface tension, although it affects the total force due to surface tension.
- Option B: Length of line considered is related to the force due to surface tension, but not the surface tension itself.
- Option C: The area of the surface and length of line considered are not direct determinants of surface tension.

Key Concepts:
- Surface Tension
- Cohesive Forces

Memory Aids:
- Surface tension is an intrinsic property of the liquid, like density or viscosity.

Quick Tips:
- Surface tension is a property of the liquid, not the geometry of the surface.

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Understanding +4 | -1

Statement A: Pascal law is working principle of a hydraulic lift.
Statement B: Pressure is equal to thrust per unit area.

View answer and explanation

Correct option: A

Statement A is true and statement B is true

Option A (Statement A is true and statement B is true) is correct because Pascal's law is indeed the working principle of a hydraulic lift, and pressure is defined as force per unit area.

Solution approach

Correct Answer + Core Reason: Statement A is true and statement B is true. Both statements are factually correct.

Step-by-Step Solution:

Step 1: Evaluate Statement A: Pascal's law is the basis for hydraulic lifts.

Step 2: Evaluate Statement B: Pressure is defined as force (thrust) per unit area.

Step 3: Both statements are true.

Why other options are wrong:
- Option B: Statement B is true, so this option is incorrect.
- Option C: Statement A is true, so this option is incorrect.
- Option D: Both statements are true, so this option is incorrect.

Key Concepts:
- Pascal's Law
- Pressure Definition

Memory Aids:
- Pascal's Law applies to hydraulic lifts.
- Pressure = Force/Area

Quick Tips:
- Remember the definitions and applications of fundamental physics principles.

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Analyzing +4 | -1

An ice block contains a glass ball when the ice melts within the water containing vessel. the level of water

View answer and explanation

Correct option: C

unchanged

Option C (unchanged) is correct because the volume of water displaced by the ice block is equal to the volume of water displaced by the glass ball after the ice melts.

Solution approach

Correct Answer + Core Reason: unchanged. The water level remains unchanged because the weight of the ice block (including the glass ball) is equal to the weight of the water it displaces. When the ice melts, the glass ball displaces its own weight in water, which is the same as the weight of the melted ice.

Step-by-Step Solution:

Step 1: Consider the ice block with the glass ball floating in water. The weight of the ice block and glass ball is equal to the buoyant force acting on it, which is equal to the weight of the water displaced.

Step 2: When the ice melts, the glass ball sinks. The glass ball now displaces a volume of water equal to its own volume.

Step 3: The melted ice becomes water. The volume of water produced by the melted ice is equal to the volume of water that the ice initially displaced minus the volume of the glass ball that is now displacing water directly.

Step 4: The total volume of water displaced after the ice melts is the volume displaced by the glass ball plus the volume of the melted ice (now water). This total volume is equal to the initial volume displaced by the ice block and glass ball.

Step 5: Since the volume of water displaced remains the same, the water level remains unchanged.

Why other options are wrong:
- Option A: The water level would rise if the volume of water displaced after melting was greater than the initial volume displaced.
- Option B: The water level would fall if the volume of water displaced after melting was less than the initial volume displaced.
- Option D: The water level does not first rise and then fall. The water level remains constant.

Key Concepts:
- Archimedes' Principle
- Buoyancy
- Density

Memory Aids:
- "What goes in must come out" - the volume displaced initially equals the volume displaced finally.

Quick Tips:
- Consider the weight of the object and the weight of the water displaced.
- Remember that the density of ice is less than the density of water, while the density of glass is greater than the density of water.

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Understanding +4 | -1

Aerofils are so designed that the speed of air

View answer and explanation

Correct option: A

On top side is more than on lower side

Option A (On top side is more than on lower side) is correct because the higher speed of air on the top creates lower pressure, generating lift according to Bernoulli's principle.

Solution approach

Correct Answer + Core Reason: On top side is more than on lower side. Higher speed on top creates lower pressure and lift.

Step-by-Step Solution:

Step 1: Recall Bernoulli's principle and its application to aerofoils.

Step 2: Bernoulli's principle states that faster-moving fluids have lower pressure.

Step 3: Aerofoils are designed so that air flows faster over the top surface than the bottom surface.

Step 4: This creates a pressure difference, with lower pressure on top and higher pressure on the bottom, resulting in an upward force called lift.

Why other options are wrong:
- Option B: If the speed were less on the top side, the pressure would be higher, resulting in a downward force.
- Option C: If the speed were the same on both sides, there would be no pressure difference and no lift.
- Option D: Turbulent flow is generally undesirable for aerofoils as it reduces lift and increases drag.

Key Concepts:
- Bernoulli's Principle
- Aerofoil
- Lift
- Fluid Dynamics

Memory Aids:
- Faster air = lower pressure.

Quick Tips:
- Understand how Bernoulli's principle explains lift.

210 Medium

210 of 218

Understanding +4 | -1

Assertion: A rain drop after falling through some height attains a constant velocity.
Reason: At constant velocity, the viscous drag is just equal to its weight.

View answer and explanation

Correct option: A

If both assertion and reason are true and reason is the correct explanation of the assertion

Option A is correct because the assertion and reason are both true, and the reason correctly explains why raindrops attain a constant velocity.

Solution approach

Correct Answer + Core Reason: If both assertion and reason are true and reason is the correct explanation of the assertion. Constant velocity is reached when drag force equals weight.

Step-by-Step Solution:

Step 1: A raindrop starts falling under the influence of gravity, which causes it to accelerate.

Step 2: As the raindrop's velocity increases, the viscous drag force (air resistance) acting on it also increases. The viscous drag force is given by Stokes' Law for a spherical object: $F = 6\pi \eta r v$ , where $\eta$ is the viscosity of the air, $r$ is the radius of the raindrop, and $v$ is its velocity.

Step 3: The viscous drag force acts upwards, opposing the downward force of gravity (weight). The weight of the raindrop is given by $W = mg$ , where $m$ is the mass of the raindrop and $g$ is the acceleration due to gravity.

Step 4: Eventually, the viscous drag force becomes equal in magnitude to the weight of the raindrop. At this point, the net force on the raindrop is zero, and it no longer accelerates. It falls with a constant velocity, called the terminal velocity.

Step 5: Therefore, the reason correctly explains why the raindrop attains a constant velocity: the viscous drag is equal to its weight.

Why other options are wrong:
- Option B: The reason *is* the correct explanation of the assertion.
- Option C: Both the assertion and the reason are true.
- Option D: Both the assertion and the reason are true.

Key Concepts:
- Terminal Velocity
- Viscous Drag
- Stokes' Law
- Weight
- Equilibrium of Forces

Memory Aids:
- At terminal velocity, Weight = Viscous Drag

Quick Tips:
- Remember that the viscous drag force depends on the velocity of the object.

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