Core Syllabus Module

Properties of Solids and Liquids

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Chapter Elite-20 Challenge Zone

Target Advanced

JEE Advanced 1998

LEVELJEE Advanced

A solid body $X$ of heat capacity $C$ is kept in an atmosphere whose temperature is $T_{A} = 300 K$ . At time $t = 0$ , the temperature of $X$ is $T_{0} = 400 K$ . It cools according to Newton's law of cooling. At time $t_{1}$ its temperature is found to be $350 K$ . At this time ( $t_{1}$ ) the body $X$ is connected to a large body $Y$ at atmospheric temperature $T_{A}$ through a conducting rod of length $L$ , cross-sectional area $A$ and thermal conductivity $K$ . The heat capacity of $Y$ is so large that any variation in its temperature may be neglected. The cross-sectional area $A$ of the connecting rod is small compared to the surface area of $X$ . Find the temperature of $X$ at time $t = 3 t_{1}$ .

JEE Advanced 2015

LEVELJEE Advanced

Two spheres $P$ and $Q$ of equal radii have densities $ρ_{1}$ and $ρ_{2}$ , respectively. The spheres are connected by a massless string and placed in liquids $L_{1}$ and $L_{2}$ of densities $σ_{1}$ and $σ_{2}$ and viscosities $η_{1}$ and $η_{2}$ , respectively. They float in equilibrium with the sphere $P$ in $L_{1}$ and sphere $Q$ in $L_{2}$ and the string being taut (see figure). If sphere $P$ alone in $L_{2}$ has terminal velocity $v_{P}$ and $Q$ alone in $L_{1}$ has terminal velocity $v_{Q}$ , then

* Multiple Correct Options

(A)

\frac{v _{P}}{v _{Q}} = \frac{η _{1}}{η _{2}}

(B)

\frac{v _{P}}{v _{Q}} = \frac{η _{2}}{η _{1}}

(C)

v_{P} \cdot v_{Q} > 0

(D)

v_{P} \cdot v_{Q} < 0

JEE Advanced 2023

LEVELJEE Advanced

Match the temperature of a black body given in List-I with an appropriate statement in List-II, and choose the correct option. [Given: Wien's constant as $2.9 \times 1 0^{- 3} m-K$ and $\frac{h c}{e} = 1.24 \times 1 0^{- 6} V-m$ ]

List-I

List-II

(1)

The radiation at peak wavelength can lead to emission of photoelectrons from a metal of work function 4 eV

(2)

The radiation at peak wavelength is visible to human eye.

(3)

The radiation at peak emission wavelength will result in the widest central maximum of a single slit diffraction.

(4)

The power emitted per unit area is 1/16 of that emitted by a blackbody at temperature 6000 K.

(5)

The radiation at peak emission wavelength can be used to image human bones.

JEE Advanced 2026

LEVELJEE Advanced

Comprehension Passage

A container of height

2 m

, length

2 m

and breadth

1 m

is made of insulating vertical walls and two large area horizontal metal plates (

M_{1}

and

M_{2}

) which extend far beyond the vertical walls in all directions. The container is partitioned into two equal chambers with a thin insulating vertical wall. The partition wall contains a small hole of cross-sectional area

10 cm^{2}

near its bottom edge. Initially the hole is closed and the left chamber of the container is completely filled with a liquid of dielectric constant

ϵ_{r} = 15

and the right chamber is empty (

ϵ_{r} = 1

). At time

t = 0

, the hole is opened and the liquid flows from the left chamber to the right chamber. In both the chambers, the space above the liquid has

ϵ_{r} = 1

and is maintained at atmospheric pressure. The schematic of the container at a time

t > 0

is shown in the figure. [Given: acceleration due to gravity is

10 ms^{- 2}

Question 1:

The height (in m) of the liquid in left chamber at $t = 500 s$ is :

Question 2:

The difference in the capacitance (in F) between the metal plates at $t = 0$ and that at $t = 500 s$ is $(8 - n) ϵ_{0}$ , where $ϵ_{0}$ is the permittivity of free space. The value of $n$ is :

JEE Advanced 2023

LEVELJEE Advanced

Comprehension Passage

A cylindrical furnace has height (H) and diameter (D) both 1 m. It is maintained at temperature 360 K. The air gets heated inside the furnace at constant pressure

P_{a}

and its temperature becomes T = 360 K. The hot air with density

ρ

rises up a vertical chimney of diameter d = 0.1 m and height h = 9 m above the furnace and exits the chimney (see the figure). As a result, atmospheric air of density

ρ_{a} = 1.2 kg m^{- 3}

, pressure

P_{a}

and temperature

T_{a} = 300 K

enters the furnace. Assume air as an ideal gas, neglect the variations in

ρ

and T inside the chimney and the furnace. Also ignore the viscous effects. [Given: The acceleration due to gravity

g = 10 ms^{- 2}

and

π = 3.14

]

Question 1:

Considering the air flow to be streamline, the steady mass flow rate of air exiting the chimney is _________ $gm s^{- 1}$ .

Question 2:

When the chimney is closed using a cap at the top, a pressure difference $Δ P$ develops between the top and the bottom surfaces of the cap. If the changes in the temperature and density of the hot air, due to the stoppage of air flow, are negligible then the value of $Δ P$ is ______ $Nm^{- 2}$ .

JEE Advanced 2022

LEVELJEE Advanced

A bubble has surface tension $S$ . The ideal gas inside the bubble has ratio of specific heats $γ = \frac{5}{3}$ . The bubble is exposed to the atmosphere and it always retains its spherical shape. When the atmospheric pressure is $P_{a 1}$ , the radius of the bubble is found to be $r_{1}$ and the temperature of the enclosed gas is $T_{1}$ . When the atmospheric pressure is $P_{a 2}$ , the radius of the bubble and the temperature of the enclosed gas are $r_{2}$ and $T_{2}$ , respectively. Which of the following statement(s) is(are) correct?

* Multiple Correct Options

(A)

If the surface of the bubble is a perfect heat insulator, then

(\frac{r _{1}}{r _{2}})^{5} = \frac{P _{a 2} + \frac{2 S}{r _{2}}}{P _{a 1} + \frac{2 S}{r _{1}}}

(B)

If the surface of the bubble is a perfect heat insulator, then the total internal energy of the bubble including its surface energy does not change with the external atmospheric pressure.

(C)

If the surface of the bubble is a perfect heat conductor and the change in atmospheric temperature is negligible, then

(\frac{r _{1}}{r _{2}})^{3} = \frac{P _{a 2} + \frac{4 S}{r _{2}}}{P _{a 1} + \frac{4 S}{r _{1}}}

(D)

If the surface of the bubble is a perfect heat insulator, then

(\frac{T _{2}}{T _{1}})^{\frac{5}{2}} = \frac{P _{a 2} + \frac{4 S}{r _{2}}}{P _{a 1} + \frac{4 S}{r _{1}}}

JEE Main 2014

LEVELJEE Advanced

On heating water, bubbles being formed at the bottom of the vessel detatch and rise. Take the bubbles to be spheres of radius $R$ and making a circular contact of radius $r$ with the bottom of the vessel. If $r ≪ R$ and the surface tension of water is $T$ , value of $r$ just before bubbles detatch is (density of water is $ρ$ )

(A)

R^{2} \frac{ρ _{w} g}{3 T}

(B)

R^{2} \frac{ρ _{w} g}{6 T}

(C)

R^{2} \frac{ρ _{w} g}{T}

(D)

R^{2} \frac{3 ρ _{w} g}{T}

JEE Main 2014

LEVELJEE Advanced

There is a circular tube in a vertical plane. Two liquids which do not mix and of densities $d_{1}$ and $d_{2}$ are filled in the tube. Each liquid subtends $9 0^{\circ}$ angle at centre. Radius joining their interface makes an angle $α$ with vertical. Ratio $\frac{d _{1}}{d _{2}}$ is

(A)

\frac{1 + s i n α}{1 - s i n α}

(B)

\frac{1 + c o s α}{1 - c o s α}

(C)

\frac{1 + t a n α}{1 - t a n α}

(D)

\frac{1 + s i n α}{1 - c o s α}

JEE Advanced 2011

LEVELJEE Advanced

A composite block is made of slabs $A, B, C, D$ and $E$ of different thermal conductivities (given in terms of a constant $K$ ) and sizes (given in terms of length, $L$ ) as shown in the figure. All slabs are of same width. Heat $Q$ flows only from left to right through the blocks. Then, in steady state

* Multiple Correct Options

(A)

heat flow through

A

and

E

slabs are same

(B)

heat flow through slab

E

is maximum

(C)

temperature difference across slab

E

is smallest

(D)

heat flow through

C

= heat flow through

B

+ heat flow through

D

JEE Advanced 2017

LEVELJEE Advanced

A human body has a surface area of approximately $1 m^{2}$ . The normal body temperature is $10 K$ above the surrounding room temperature $T_{0}$ . Take the room temperature to be $T_{0} = 300 K$ . For $T_{0} = 300 K$ , the value of $σ T_{0}^{4} = 460 Wm^{- 2}$ (where $σ$ is the Stefan Boltzmann constant). Which of the following options is/are correct?

* Multiple Correct Options

(A)

If the body temperature rises significantly, then the peak in the spectrum of electromagnetic radiation emitted by the body would shift to longer wavelengths

(B)

If the surrounding temperature reduces by a small amount

Δ T_{0} ≪ T_{0}

, then to maintain the same body temperature the same (living) human being needs to radiate

Δ W = 4 σ T_{0}^{3} Δ T_{0}

more energy per unit time

(C)

The amount of energy radiated by the body in 1s is close to 60 J

(D)

Reducing the exposed surface area of the body (e.g. by curling up) allows humans to maintain the same body temperature while reducing the energy lost by radiation

LEVELJEE Advanced

The apparatus shown in figure consists of four glass columns connected by horizontal sections. The height of two central columns $B$ and $C$ are $49 cm$ each. The two outer columns $A$ and $D$ are open to the atmosphere. $A$ and $C$ are maintained at a temperature of $9 5^{\circ} C$ while the columns $B$ and $D$ are maintained at $5^{\circ} C$ . The height of the liquid in $A$ and $D$ measured from the base line are $52.8 cm$ and $51 cm$ respectively. Determine the linear coefficient of thermal expansion of the liquid.

JEE Advanced 2004

LEVELJEE Advanced

One end of a rod of length $L$ and cross-sectional area $A$ is kept in a furnace of temperature $T_{1}$ . The other end of the rod is kept at a temperature $T_{2}$ . The thermal conductivity of the material of the rod is $K$ and emissivity of the rod is $e$ . It is given that $T_{2} = T_{s} + Δ T$ , where $Δ T ≪ T_{s}$ , $T_{s}$ being the temperature of the surroundings. If $Δ T \propto (T_{1} - T_{s})$ , find the proportionality constant. Consider that heat is lost only by radiation at the end where the temperature of the rod is $T_{2}$ .

LEVELJEE Advanced

A 5 m long cylindrical steel wire with radius $2 \times 1 0^{- 3} m$ is suspended vertically from a rigid support and carries a bob of mass 100 kg at the other end. If the bob gets snapped, calculate the change in temperature of the wire ignoring losses. (For the steel wire : Young's modulus $= 2.1 \times 1 0^{11} Pa$ ; Density $= 7860 kg/m^{3}$ ; Specific heat $= 420 J/kg-K$ ).

LEVELJEE Advanced

A cylindrical block of length $0.4 m$ and area of cross-section $0.04 m^{2}$ is placed coaxially on a thin metal disc of mass $0.4 kg$ and of the same cross-section. The upper face of the cylinder is maintained at a constant temperature of $400 K$ and the initial temperature of the disc is $300 K$ . If the thermal conductivity of the material of the cylinder is $10 W/mK$ and the specific heat capacity of the material of the disc is $600 J/kg-K$ , how long will it take for the temperature of the disc to increase to $350 K$ ? Assume, for purposes of calculation, the thermal conductivity of the disc to be very high and the system to be thermally insulated except for the upper face of the cylinder.

JEE Advanced 2016

LEVELJEE Advanced

The ends $Q$ and $R$ of two thin wires, $P Q$ and $R S$ , are soldered (joined) together. Initially, each of the wire has a length of $1 m$ at $1 0^{\circ} C$ . Now, the end $P$ is maintained at $1 0^{\circ} C$ , while the end $S$ is heated and maintained at $40 0^{\circ} C$ . The system is thermally insulated from its surroundings. If the thermal conductivity of wire $P Q$ is twice that of the wire $R S$ and the coefficient of linear thermal expansion of $P Q$ is $1.2 \times 1 0^{- 5} K^{- 1}$ , the change in length of the wire $P Q$ is

A sinker of weight $w_{0}$ has an apparent weight $w_{1}$ when placed in a liquid at a temperature $T_{1}$ and $w_{2}$ when weighed in the same liquid at a temperature $T_{2}$ . The coefficient of cubical expansion of the material of the sinker is $β$ . What is the coefficient of volume expansion of the liquid ?

JEE Advanced 2003

LEVELJEE Advanced

A liquid of density $900 kg/m^{3}$ is filled in a cylindrical tank of upper radius $0.9 m$ and lower radius $0.3 m$ . A capillary tube of length $l$ is attached at the bottom of the tank as shown in the figure. The capillary has outer radius $0.002 m$ and inner radius $a$ . When pressure $p$ is applied at the top of the tank volume flow rate of the liquid is $8 \times 1 0^{- 6} m^{3} /s$ and if capillary tube is detached, the liquid comes out from the tank with a velocity $10 m/s$ . Determine the coefficient of viscosity of the liquid. [Given, $π a^{2} = 1 0^{- 6} m^{2}$ and $a^{2} / l = 2 \times 1 0^{- 6} m$ ]

JEE Advanced 1997

LEVELJEE Advanced

A large open top container of negligible mass and uniform cross-sectional area $A$ has a small hole of cross-sectional area $A /100$ in its side wall near the bottom. The container is kept on a smooth horizontal floor and contains a liquid of density $ρ$ and mass $m_{0}$ . Assuming that the liquid starts flowing out horizontally through the hole at $t = 0$ . Calculate (a) the acceleration of the container and (b) velocity of efflux when 75% of the liquid has drained out.

JEE Advanced 2003

LEVELJEE Advanced

A soap bubble is being blown at the end of very narrow tube of radius $b$ . Air (density $ρ$ ) moves with a velocity $v$ inside the tube and comes to rest inside the bubble. The surface tension of the soap solution is $T$ . After sometime the bubble, having grown to radius $r$ separates from the tube. Find the value of $r$ . Assume that $r ≫ b$ so, that you can consider the air to be falling normally on the bubble's surface.

JEE Advanced 1995

LEVELJEE Advanced

A container of large uniform cross-sectional area $A$ resting on a horizontal surface, holds two immiscible, non-viscous and incompressible liquids of densities $d$ and $2 d$ , each of height $H /2$ as shown in figure. The lower density liquid is open to the atmosphere having pressure $p_{0}$ . (a) A homogeneous solid cylinder of length $L$ ( $L < H /2$ ), cross-sectional area $A /5$ is immersed such that it floats with its axis vertical at the liquid-liquid interface with length $L /4$ in the denser liquid. Determine (i) the density $D$ of the solid, (ii) the total pressure at the bottom of the container. (b) The cylinder is now removed and the original arrangement is restored. A tiny hole of area $s$ ( $s ≪ A$ ) is punched on the vertical side of the container at a height $h$ ( $h < H /2$ ). Determine: (i) the initial speed of efflux of the liquid at the hole, (ii) the horizontal distance $x$ travelled by the liquid initially, and (iii) the height $h_{m}$ at which the hole should be punched so that the liquid travels the maximum distance $x_{m}$ initially. Also calculate $x_{m}$ . (Neglect the air resistance in these calculations)

Micro-Topic Strategy Matrix

Young's Modulus, Bulk Modulus and Modulus of Rigidity

Fluid Pressure and Pascal's Law

Buoyancy and Archimedes' Principle

Viscosity and Stokes' Law

Flow of Fluid

Surface Tension and Capillary Action

Thermal Expansion

Calorimetry

Heat Transfer

Chapter Elite-20 Challenge Zone

List-I

List-II

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The height (in m) of the liquid in left chamber at t=500 s is :

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Considering the air flow to be streamline, the steady mass flow rate of air exiting the chimney is _________ gm s−1.

The height (in m) of the liquid in left chamber at $t = 500 s$ is :

Considering the air flow to be streamline, the steady mass flow rate of air exiting the chimney is _________ $gm s^{- 1}$ .