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Calorimetry Thermodynamics

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Description

BANSALCLASSES TARGET IIT JEE

PHYSICS BULLS EYE

CALORIMETRY & HEAT TRANSFER

CONTENTS

KEY CONCEPT EXERCISE–I EXERCISE–II EXERCISE–III ANSWER KEY

THERMAL

EXPANSION

Definition of Heat : Heat is a form of energy which is transferred between a system and its surrounding as a result of temperature difference only

Thermal Expansion : Expansion due to increase in temperature

Type of thermal expansion

Coefficient of expansion

For temperature change t change in length l = l0 t Area A = A0t volume V = V0t

For isotropic solids 1 = 2 = 3 =  (let) so =2 and  = 3

For anisotropic solids =1 + 2 and  = 1 + 2 + 3 Here 1 ,

2 and 3 are coefficient of linear expansion in X ,

Y and Z directions

Variation in density : With increase of temperature volume increases so density decreases and vice-versa

Note : (i)  for liquids are in order of 10–3 (ii) For water density increases from 0 to 4°C so  is –ve (0 to 4° C) and for 4° C to higher temperature  is +ve

At 4° C density is maximum

Thermal Stress : A rod of length l0 is clamped between two fixed walls with distance l0

If temperature is changed by amount t then F stress = (area assumed to be constant) A strain = 

FA Fl 0 F Y = l l'= = At Al 0 F = Y A t

Bansal Classes

Calorimetry & Heat Transfer

If  is not constant

( varies with distance) Let  = ax+b

Total expansion =  expansion of length dx =  (ax  b)dx t 0

( varies with tempearture) Let  = f (T) l =

Caution : If is in °C then put T1 and T2 in °C

similarly if is in K then put T1 and T2 in K

CALORIMETRY

Quantity of heat transfered and specific heat

The amount of heat needed to incerase the temperature of 1 gm of water from 14

5° C to 15

5°C at STP is 1 calorie dQ = mcdT Q = m  C dT (be careful about unit of temperature,

use units according to the given units of C) T2 T1

Heat transfer in phase change Q = mL L'= latent heat of substance in cal/ gm/ °C or in Kcal/ kg/ °C Lice = 80 cal/ gm for ice L'steam = 540 cal/ gm (A) (i)

- TRANSFER

Conduction : Due to vibration and collision of medium particles

Steady State : In this state heat absorption stops and temperature gradient throughout the rod dT becomes constant i

dx Before steady state : Temp of rod at any point changes

Note : If specific heat of any substance is zero,

it can be considered always in steady state

Ohm’s law for Thermal Conduction in Steady State : Let the two ends of rod of length l'is maintained at temp T1 and T2( T1 > T2) dQ T1  T2 Thermal current = R dT Th Where thermal resistance RTh =

Differential form of Ohm’s Law dQ dT = KA dT dx

Bansal Classes

dT = temperature gradient dx

Calorimetry & Heat Transfer

Convection : Heat transfer due to movement of medium particles

Stefan’s Law : Rate of heat emitted by a body at temp T K from per unit area E =T4 J/sec/m2 dQ Radiation power = P = AT T4 watt dT If a body is placed in a surrounding of temperature TS dQ = A (T4 – Ts4) dT valid only for black body

Radiation: Every body radiates electromagnetic radiation of all possible wavelength at all temp>0 K

heat from general body Emissivity or emmisive power e = heat from black body If temp of body falls by dT in time dt dT eA 4  (T  Ts4 ) (dT/dt = rate of cooling) dt mS

Newton’s law of cooling dT  (T  Ts ) dt

Average form of Newtons law of cooling If a body cools from T1 to T2 in time  t

T1  T2  K  T1  T2    Ts  (used generally in objective questions) t mS  2  dT K  (T  Ts ) dt mS

(for better results use this generally in subjective)

Wein’s black body radiation At every temperature (>0K) a body radiates energy radiations of all wavelengths

According to Wein’s displacement law if the wavelength corresponding to maximum energy is m

then mT = b where b = is a constant (Wein’s constant) T = temperature of body

Bansal Classes

Calorimetry & Heat Transfer

EXERCISE – I

A light rigid bar is suspended horizontally from two vertical wires,

one of steel and one of brass,

Each wire is 2

00 m long

The diameter of the steel wire is 0

When a mass of 10 kg is suspended from the centre of AB bar remains horizontal

What is the tension in each wire

? Calculate the extension of the steel wire and the energy stored in it

Calculate the diameter of the brass wire

If the brass wire were replaced by another brass wire of diameter 1 mm,

where should the mass be suspended so that AB would remain horizontal

? The Young modulus for steel = 2

the Young modulus for brass = 1

A steel rope his length L,

Young’s modulus Y

[Density = d] It is pulled on a horizontal frictionless floor with a constant horizontal force F = [dALg]/2 applied at one end

Find the strain at the midpoint

If the steel rope is vertical and moving with the force acting vertically up at the upper end

Find the strain at a point L/3 from lower end

An aluminium container of mass 100 gm contains 200 gm of ice at – 20°C

Heat is added to the system at the rate of 100 cal/s

Find the temperature of the system after 4 minutes (specific heat of ice = 0

A U-tube filled with a liquid of volumetric coefficient of 10–5/°C lies in a vertical plane

The height of liquid column in the left vertical limb is 100 cm

The liquid in the left vertical limb is maintained at a temperature = 0°C while the liquid in the right limb is maintained at a temperature = 100°C

Find the difference in levels in the two limbs

A thin walled metal tank of surface area 5m2 is filled with water tank and contains an immersion heater dissipating 1 kW

The tank is covered with 4 cm thick layer of insulation whose thermal conductivity is 0

2 W/m/K

The outer face of the insulation is 25oC

Find the temperature of the tank in the steady state

A glass flask contains some mercury at room temperature

It is found that at different temperatures the volume of air inside the flask remains the same

If the volume of mercury in the flask is 300 cm3,

then find volume of the flask (given that coefficient of volume expansion of mercury and coefficient of linear expansion of glass are 1

A clock pendulum made of invar has a period of 0

If the clock is used in a climate where average temperature is 30°C,

How much fast or slow will the clock run in 106 sec

(invar=1×10–6/°C) A pan filled with hot food cools from 50

How long will it take to cool from 40

9°C if room temperature is 30°C

A composite rod made of three rods of equal length and cross-section as shown in the fig

The thermal conductivities of the materials of the rods are K/2,

5K and K respectively

The end A and end B are at constant temperatures

All heat entering the face A goes out of the end B there being no loss of heat from the sides of the bar

Find the effective thermal conductivity of the bar A

An iron bar (Young’s modulus = 1011 N/m2 ,

 = 10–6 /°C) 1 m long and 10–3 m2 in area is heated from

0°C to 100°C without being allowed to bend or expand

Find the compressive force developed inside the bar

Bansal Classes

Calorimetry & Heat Transfer

A solid copper cube and sphere,

both of same mass & emissivity are heated to same initial temperature and kept under identical conditions

What is the ratio of their initial rate of fall of temperature

A cylindrical rod with one end in a stream chamber and other end in ice cause melting of 0

If the rod is replaced with another rod of half the length and double the radius of first and thermal conductivity of second rod is 1/4 that of first,

find the rate of ice melting in gm/sec Three aluminium rods of equal length form an equilateral triangle ABC

Taking O (mid point of rod BC) as the origin

Find the increase in Y-coordinate of center of mass per unit change in temperature of the system

Assume the length of the each rod is 2m,

and al = 4 3  10 6 / C

Three conducting rods of same material and cross-section are shown in figure

Temperature of A,

D and C are maintained at 20°C,

Find the ratio of length BD and BC if there is no heat flow in AB

If two rods of length L'and 2 L'having coefficients of linear expansion  and 2 respectively are connected so that total length becomes 3 L,

determine the average coefficient of linear expansion of the composite rod

A volume of 120 ml of drink (half alcohol + half water by mass) originally at a temperature of 25°C is cooled by adding 20 gm ice at 0°C

If all the ice melts,

find the final temperature of the drink

833 gm/cc,

A solid receives heat by radiation over its surface at the rate of 4 kW

The heat convection rate from the surface of solid to the surrounding is 5

and heat is generated at a rate of 1

The rate of change of the average temperature of the solid is 0

5°Cs–1

Find the heat capacity of the solid

The figure shows the face and interface temperature of a composite slab containing of four layers of two materials having identical thickness

Under steady state condition,

find the value of temperature 

Two identical calorimeter A and B contain equal quantity of water at 20°C

A 5 gm piece of metal X of specific heat 0

The equilibrium temperature in A is 22°C and in B 23°C

The initial temperature of both the metals is 40°C

Find the specific heat of metal Y in cal g–1 (C°)–1

Two spheres of same radius R have their densities in the ratio 8 : 1 and the ratio of their specific heats are 1 : 4

If by radiation their rates of fall of temperature are same,

then find the ratio of their rates of losing heat

In the square frame of side l'of metallic rods,

the corners A and C are maintained at T1 and T2 respectively

The rate of heat flow from A to C is 

If A and D'are instead maintained T1 & T2 respectivley find,

find the total rate of heat flow

A hot liquid contained in a container of negligible heat capacity loses temperature at rate 3 K/min,

just before it begins to solidify

The temperature remains constant for 30 min

Find the ratio of specific heat capacity of liquid to specific latent heat of fusion is in K–1 (given that rate of losing heat is constant)

Bansal Classes

Calorimetry & Heat Transfer

A thermostatted chamber at small height h above earth's surface maintained at 30°C has a clock fitted in it with an uncompensated pendulum

The clock designer correctly designs it for height h,

If this chamber is taken to earth's surface,

the clock in it would click correct time

Find the coefficient of linear expansion of material of pendulum

(earth's radius is R) The coefficient of volume expansion of mercury is 20 times the coefficient of linear expansion of glass

Find the volume of mercury that must be poured into a glass vessel of volume V so that the volume above mercury may remain constant at all temperature

Two 50 gm ice cubes are dropped into 250 gm of water into a glass

If the water was initially at a temperature of 25°C and the temperature of ice –15°C

Find the final temperature of water

Find final amount of water and ice

Water is heated from 10°C to 90°C in a residential hot water heater at a rate of 70 litre per minute

Natural gas with a density of 1

which has a transfer efficiency of 32%

Find the gas consumption rate in cubic meters per hour

(heat combustion for natural gas is 8400 kcal/kg) A metal rod A of 25cm lengths expands by 0

When its temperature is raised from 0°C to 100°C

Another rod B of a different metal of length 40cm expands by 0

A third rod C of 50cm length is made up of pieces of rods A and B placed end to end expands by 0

Find the lengths of each portion of the composite rod

A substance is in the solid form at 0°C

The amount of heat added to this substance and its temperature are plotted in the following graph

If the relative specific heat capacity of the solid substance is 0

find from the graph the mass of the substance

the specific latent heat of the melting process,

and the specific heat of the substance in the liquid state

One end of copper rod of uniform cross-section and of length 1

At what point along its length should a temperature of 200°C be maintained,

the mass of ice melting is equal to that of steam produced in the same interval of time

? Assume that the whole system is insulated from the surroundings

A vessel containing 100 gm water at 0°C is suspended in the middle of a room

In 15 minutes the temperature of the water rises by 2°C

When an equal amount of ice is placed in the vessel,

Calculate the specific heat of fusion of ice

The maximum in the energy distribution spectrum of the sun is at 4753 Å and its temperature is 6050K

What will be the temperature of the star whose energy distribution shows a maximum at 9506 Å

List of recommended questions from I

247 to 2

254 to 2

Bansal Classes

Calorimetry & Heat Transfer

EXERCISE – II

A wire of cross-secitonal area 4 × 10–4 m2 modulus of elasticity 2 × 1011 N/m2 and length 1 m is stretched between two vertical rigid poles

A mass of 1 kg is suspended at its middle

Calculate the angle it makes with the horizontal

A metal spherical shell of mean radius 20 cm and wall thickness 1 mm is completely filled with a liquid of bulk modulus = 10 GPa,

and volumetric thermal expansion coefficient = 2 × 10–4 /K

Then shell is then sealed from all sides at room temperature,

with the liquid at atmospheric pressure

The Young’s modulus of the metal shell is 100 GPa and its linear thermal expansion coefficient is 1 × 10–5 /K

The temperature of the system is now raised by T°C

If the fracture tensile stress of the metal shell is 0

at what value of T will it rupture

A copper calorimeter of mass 100 gm contains 200 gm of a mixture of ice and water

Steam at 100°C under normal pressure is passed into the calorimeter and the temperature of the mixture is allowed to rise to 50°C

If the mass of the calorimeter and its contents is now 330 gm,

what was the ratio of ice and water in the beginning

Given : Specific heat capacity of copper = 0

Specific heat capacity of water = 4

Specific heat of fusion of ice = 3

An isosceles triangle is formed with a rod of length l1 and coefficient of linear expansion 1 for the base and two thin rods each of length l2 and coefficient of linear expansion 2 for the two pieces,

if the distance between the apex and the midpoint of the base remain unchanged as the temperatures varied show that

A solid substance of mass 10 gm at – 10°C was heated to – 2°C (still in the solid state)

The heat required was 64 calories

Another 880 calories was required to raise the temperature of the substance (now in the liquid state) to 1°C,

while 900 calories was required to raise the temperature from –2°C to 3°C

Calculate the specific heat capacities of the substances in the solid and liquid state in calories per kilogram per kelvin

Show that the latent heat of fusion L'is related to the melting point temperature tm by L'= 85400 + 200 tm

A steel drill making 180 rpm is used to drill a hole in a block of steel

The mass of the steel block and the drill is 180 gm

If the entire mechanical work is used up in producing heat and the rate of raise in temperature of the block and the drill is 0

5 °C/s

Find the rate of working of the drill in watts,

and the torque required to drive the drill

Specific heat of steel = 0

2 J/cal

Use : P =  A brass rod of mass m = 4

What amount of heat is spent for heating the rod

? The coefficient of linear expansion for brass is 2×10–5/K,

39 kJ/kg

K and the density of brass is 8

Bansal Classes

Calorimetry & Heat Transfer

A submarine made of steel weighing 109 g has to take 108 g of water in order to submerge when the temperature of the sea is 10°C

How much less water it will have to take in when the sea is at 15°C

? (Coefficient of cubic expansion of sea water = 2 × 10–4/°C,

coefficient of linear expansion of steel = 1

A flow calorimeter is used to measure the specific heat of a liquid

Heat is added at a known rate to a stream of the liquid as it passes through the calorimeter at a known rate

Then a measurement of the resulting temperature difference between the inflow and the outflow points of the liquid stream enables us to compute the specific heat of the liquid

A liquid of density 0

Heat is added by means of a 250-W electric heating coil,

and a temperature difference of 25°C is established in steady-state conditions between the inflow and the outflow points

Find the specific heat of the liquid

Toluene liquid of volume 300 cm3 at 0°C is contained in a beaker an another quantity of toluene of volume 110 cm3 at 100°C is in another beaker

(The combined volume is 410 cm3)

Determine the total volume of the mixture of the toluene liquids when they are mixed together

Given the coefficient of volume expansion  = 0

Also find the final temperature of the mixture

Water at temperature °C is filled in another identical vessel upto the same height h= 10 cm

water from second vessel is poured into first vessel and it is found that level of upper surface falls through h = 0

Neglecting thermal capacity of vessels,

change in density of water due to change in temperature and loss of heat due to radiation,

calculate initial temperature  of water

Density of water,

w = 1 gm cm–3 Density of ice,

sw = 1 cal/gm 0C Specific heat of ice,

L = 80 cal/gm

A composite body consists of two rectangular plates of the same dimensions but different thermal conductivities KA and KB

This body is used to transfer heat between two objects maintained at different temperatures

The composite body can be placed such that flow of heat takes place either parallel to the interface or perpendicular to it

Calculate the effective thermal conductivities K| | and

K  of the composite body for the parallel and perpendicular orientations

Which orientation will have more thermal conductivity

A highly conducting solid cylinder of radius a and length l'is surrounded by a co-axial layer of a material having thermal conductivity K and negligible heat capacity

Temperature of surrounding space (out side the layer) is T0,

which is higher than temperature of the cylinder

If heat capacity per unit volume of cylinder material is s'and outer radius of the layer is b,

calculate time required to increase temperature of the cylinder from T1 to T2

Assume end faces to be thermally insulated

A vertical brick duct(tube) is filled with cast iron

The lower end of the duct is maintained at a temperature T1 which is greater than the melting point Tm of cast iron and the upper end at a temperature T2 which is less than the temperature of the melting point of cast iron

It is given that the conductivity of liquid cast iron is equal to k times the conductivity of solid cast iron

Determine the fraction of the duct filled with molten metal

Bansal Classes

Calorimetry & Heat Transfer

A lagged stick of cross section area 1 cm2 and length 1 m is initially at a temperature of 0°C

It is then kept between 2 reservoirs of tempeature 100°C and 0°C

Specific heat capacity is 10 J/kg°C and linear mass density is 2 kg/m

temperature gradient along the rod in steady state

total heat absorbed by the rod to reach steady state

A cylindrical block of length 0

The upper face of the cylinder is maintained at a constant temperature of 400K and the initial temperature of the disc is 300K

If the thermal conductivity of the material of the cylinder is 10 watt/m-K and the specific heat of the material of the disc in 600 J/kgK,

how long will it take for the temperature of the disc to increase to 350K

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

A solid copper sphere cools at the rate of 2

8°C per minute,

when its temperature is 127°C

Find the rate at which another solid copper sphere of twice the radius lose its temperature at 327°C,

the room temperature is maintained at 27°C

End A of a rod AB of length L'= 0

The heat conductivity of the rod is k = 17 J/s-m°K

The other end B of this rod is radiating energy into vacuum and the wavelength with maximum energy density emitted from this end is 0 = 75000 Å

If the emissivity of the end B is e = 1,

determine the temperature of the end A

Assuming that except the ends,

the rod is thermally insulated

The shell of a space station is a blackened sphere in which a temperature T = 500K is maintained due to operation of appliances of the station

Find the temperature of the shell if the station is enveloped by a thin spherical black screen of nearly the same radius as the radius of the shell

A liquid takes 5 minutes to cool from 80°C to 50°C

How much time will it take to cool from 60°C to 30°C

? The temperature of surrounding is 20°C

Use exact method

Bansal Classes

Calorimetry & Heat Transfer

EXERCISE – III

The temperature of 100 gm of water is to be raised from 24º C to 90º C by adding steam to it

Calculate the mass of the steam required for this purpose

Two metal cubes A & B of same size are arranged as shown in figure

The extreme ends of the combination are maintained at the indicated temperatures

The arrangement is thermally insulated

The coefficients of thermal conductivity of A & B are 300 W/mº C and 200 W/mº C respectively

After steady state is reached the temperature T of the interface will be ______

A double pane window used for insulating a room thermally from outside consists of two glass sheets each of area 1 m2 and thickness 0

In the steady state,

the room glass interface and the glass outdoor interface are at constant temperatures of 270C and 00C respectively

Calculate the rate of heat flow through the window pane

Also find the temperatures of other interfaces

Given thermal conductivities of glass and air as 0

8 and 0

The apparatus shown in the figure consists of four glass columns connected by horizontal sections

The height of two central columns B & C are 49 cm each

The two outer columns A & D'are open to the atmosphere

A & C are maintained at a temperature of 95º C while the columns B & D'are maintained at 5º C

The height of the liquid in A & D'measured from the base line are 52

Determine the coefficient of thermal expansion of the liquid

A spherical black body with a radius of 12 cm radiates 450 W power at 500 K

If the radius were halved and the temperature doubled,

the power radiated in watt would be : (A) 225 (B) 450 (C) 900 (D) 1800

Earth receives 1400 W/m2 of solar power

If all the solar energy falling on a lens of area 0

the time taken to melt the ice will be ______ minutes

(Latent heat of fusion of ice = 3

A solid body X of heat capacity C is kept in an atmosphere whose temperature is TA = 300K

At time t = 0,

the temperature of X is T0 = 400K

It cools according to Newton’s law of cooling

At time t1 its temperature is found to be 350K

At this time t1,

the body X is connected to a larger body Y at atmospheric temperature TA,

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 = 3t1

A black body is at a temperature of 2880 K

The energy of radiation emitted by this object with wavelength between 499 nm and 500 nm is U1,

between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3

The Wien constant b = 2

Then [JEE’ 98] (A) U1 = 0 (B) U3 = 0 (C) U1 > U2 (D) U2 > U1

Bansal Classes

Calorimetry & Heat Transfer

A bimetallic strip is formed out of two identical strips one of copper and the other of brass

The coefficient of linear expansion of the two metals are C and B

On heating,

the temperature of the strip goes up by T and the strip bends to form an arc of radius of curvature R

Then R is : (A) proportional at T (B) inversely proportional to T [JEE’ 99] (C) proportional to |B – C| (D) inversely proportional to |B – C| A block of ice at – 10°C is slowly heated and converted to steam at 100°C

Which of the following curves represents the phenomenon qualitatively

The plots of intensity versus wavelength for three black bodies at temperature T1 ,

T2 and T3 respectively are as shown

Their temperatures are such that [JEE (Scr) 2000] (A) T1 > T2 > T3 (B) T1 > T3 > T2 (C) T2 > T3 > T1 (C) T3 > T2 > T1

Three rods made of the same material and having the same cross-section have been joined as shown in the figure

Each rod is of the same length

The left and right ends are kept at 0°C and 90°C respectively

The temperature of the junction of the three rods will be [JEE(Scr)2001] (A) 45°C (B) 60°C (C) 30°C (D) 20°C

An ideal black body at room temperature is thrown into a furnace

It is observed that (A) initially it is the darkest body and at later times the brightest

(B) it the darkest body at all times (C) it cannot be distinguished at all times

(D) initially it is the darkest body and at later times it cannot be distinguished

An ice cube of mass 0

The specific heat S of the container varies with temperature T according the empirical relations = A + BT,

where A = 100 cal/kg-K and B = 2 × 10–2 cal/kg-K2

If the final temperature of the container is 27°C,

determine the mass of the container

(Latent heat of fusion for water = 8 × 104 cal/kg

Specific heat of water = 103 cal/kg-K) [JEE' 2001]

Two rods one of aluminium of length l1 having coefficient of linear expansion a,

and other steel of length l2 having coefficient of linear expansion s are joined end to end

The expansion in both the l1 rods is same on variation of temperature

Then the value of l' l'is 1 2 s (A)    a s

Bansal Classes

Calorimetry & Heat Transfer

[JEE' (Scr) 2003]

(D) None of these

Then final amount of water in the mixture would be

Given specific heat of ice = 0

5cal/g°C,

specific heat of water = 1 cal/g°C,

Latent heat of fusion of ice = 80 cal/g

[JEE' (Scr) 2003] (A) 6 kg (B) 5 kg (C) 4 kg (D) 2 kg If emissivity of bodies X and Y are ex and ey and absorptive power are Ax and Ay then [JEE' (Scr) 2003] (A) ey > ex

Ay > Ax (B) ey < ex

Ay < Ax (C) ey > ex

Ay < Ax (D) ey = ex

Ay = Ax

Hot oil is circulated through an insulated container with a wooden lid at the top whose conductivity K = 0

Temperature of the top of the lid in steady state is at Tl = 127°

If the ambient temperature Ta = 27°C

Calculate rate of heat loss per unit area due to radiation from the lid

Three discs A,

The wavelengths corresponding to maximum intensity are 300 nm,

The power radiated by them are QA,

QB and QC respectively

(a) QA is maximum (B) QB is maximum [JEE' 2004 (Scr

)] (C) QC is maximum (D) QA = QB = QC

Two identical conducting rods are first connected independently to two vessels,

one containing water at 100°C and the other containing ice at 0° C

In the second case,

the rods are joined end to end and connected to the same vessels

Let q1 and q2 g/s be the rate of melting of ice in the two cases respectively

The ratio q2/q1 is (A) 1/2 (B) 2/1 (C) 4/1 (D) 1/4 [JEE' 2004 (Scr

)] Liquid oxygen at 50 K is heated to 300 K at constant pressure of 1 atm

The rate of heating is constant

Which of the following graphs represents the variation of temperature with time

[JEE' 2003]

[JEE' 2004 (Scr

)] A cube of coefficient of linear expansion s is floating in a bath containing a liquid of coefficient of volume expansion l

When the temperature is raised by T,

the depth upto which the cube is submerged in the liquid remains the same

Find the relation between s and l,

[JEE 2004] One end of a rod of length L'and cross-sectional area A is kept in a furnace of temperature T1

The other end of the rod is kept at a temperature T2

The thermal conductivity of the material of the rod is K and emissivity of the rod is e

It is given that T2 = TS + T where T