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Heat and Temperature [ + a(3 C)] IQ 75 cm- [+ Y(3 C)] - 75 cm [ + (a - J) (3 C)] - 74-89 cm. The lenth of mercury column at 3 C is l. Suppose the lenth of the mercury column, if it were at C, is l.

1 Heat and Temperature [ + a(3 C)] IQ 75 cm- [+ Y(3 C)] - 75 cm [ + (a - J) (3 C)] cm. The lenth of mercury column at 3 C is l. Suppose the lenth of the mercury column, if it were at C, is l. Then, le-u +^Y(3 C)].... GO By (i) and (ii), U + y (3 C)] - 75 cm[l * a(3 C)] QUESTIONS FOR SHORT ANSWER. If two bodies are in thermal equilibrium in one frame, will they be in thermal equilibrium in all frames? 2. Does the temperature of a body depend on the frame from which it is observed? 3. It is heard sometimes that mercury is used in definin the temperature scale because it expands uniformly with the temperature. If the temperature scale is not yet defined, is it loical to say that a substance expands uniformly with the temperature? 4. In definin the ideal as temperature scale, it is assumed that the pressure of the as at constant volume is proportional to the temperature T. How can we verify whether this is ue or not? Are we usin the kinetic theory of ases? Are we usin the experimental result that the pressure is proportional to temperature? 5. Can the bulb of a thermometer be made of an adiabatic wall? 6. Why do marine animals live deep inside a lake when the surface of the lake freezes? 7. The lenth of a brass rod is found to be smaller on a hot summer day than on a cold winter day as measured by the same aluminium scale. Do we conclude that brass shrinks on heatin? 8. If mercury and lass had equal coefficient of volume expansion, could we make a mercury thermometer in a lass tube? 9. The density of water at 4 C is supposed to be k/m 3. Is it same at the sea level and at a hih altitude?. A tihtly closed metal lid of a lass bottle can be opened more easily if it is put in hot water for some time. Explain.. If an autompbile enine is overheated, it is cooled by puttin water on it. It is advised that the water should be put slowly with enine runnin. Explain the reason. 2. Is it possible for two bodies to be in thermal equilibrium if they are not. in contact? 3. A spherical shell is heated. The volume chanes accordin to the equation Ve V ( + y). Does the volume refer to the volume enclosed by the shell or the volume of the material makin up the shell? OBJECTIVE I. A system X is neither in thermal equilibrium with Y nor with Z. The systems Y and Z (a) must be in thermal equilibrium (b) cannot be in thermal equilibrium (c) may be in thermal equilibrium. 2. Which of the curves in fiure (23-Ql) represents the relation between Celsius and Fahrenheit temperatures? c Fiure 23-Ql 3. Which of the followin pairs may ive equal numerical values of the temperature of a body? (a) Fahrenheit and kelvin (b) Celsius and kelvin (c) kelvin and platinum. 4. For a constant volume as thermometer, one should fill the as at (a) low temperature and low pressure (b) low temperature and hih pressure (c) hih temperature and low pressure (d) hih temperature and hih pressure. 5. Consider the followin statements. (A) The coefficient of linear expansion has dimension K-. (B) The coefficient of volume expansion has dimension K'\ (a) A and B are both correct. (b) A is correct but B is wron.

2 2 Concepts of hysics (c) B is correct but A is wron. (d) A and B are both wron. 6. A metal sheet with a circular hole is heated. The hole (a) ets larer (b) ets smaller (c) remains of the same size (d) ets deformed. 7. Two identical rectanular sips, one of copper and the other of steel, are rivetted toether to form a bimetallic sip (a^ppe,. > a8teci). On heatin, this sip will (a) remain saiht (b) bend with copper on convex side (c) bend with steel on convex side (d) et twisted. 8. If the temperature of a uniform rod is slihtly increased by At, its moment of inertia I about a perpendicular bisector increases by (a) zero (b) cdat (c) 2odAt (d) 3alAt. 9. If the temperature of a uniform rod is slihtly increased by At, its moment of inertia I about a line parallel to itself will increase by (a) zero (b) alat (c) 2alAt (d) SalAt.. The temperature of water at the surface of a deep lake is 2 C. The temperature expected at the bottom is (a) C (b) 2 C (c) 4 C (d) G C.. An aluminium sphere is dipped into water at C. If the temperature is increased, the force of buoyancy (a) will increase (b) will decrease (c) will remain constant (d) may increase or decrease dependin on the radius of the sphere. OBJECTIVE II. A spinnin wheel is brouht in contact with an identical wheel spinnin at identical speed. The wheels slow down under the action of friction. Which of the followin eneries of the first wheel decrease? (a), kinetic (b) total (c) mechanical (d) internal. 2. A spinnin wheel A is brouht in contact with another wheel E initially at rest. Because of the friction at contact, the second wheel also starts spinnin. Which of the followin eneries of the wheel B increase? (a) kinetic (b) total (c) mechanical (d) internal. 3. A body A is placed on a railway platform and an identical body B in a movin ain. Which of the followin eneries of B are reater than those of A as seen from the round? (a) kinetic (b) total (c) mechanical (d) internal. 4. In which of the followin pairs of temperature scales, the size of a deree is identical? (a) mercury scale and ideal as scale (b) Celsius scale and mercury scale (c) Celsius scale and ideal as scale (d) ideal as scale and absolute scale. 5. A solid object is placed in water contained in an adiabatic container for some time. The temperature of water falls durin the period and there is no appreciable chane in the shape of the object. The temperature of the solid object (a) must have increased (b) must have decreased (c) may have increased (d) may have remained constant. 6. As the temperature is increased, the time period of a pendulum (a) increases proportionately with temperature (b) increases (c) decreases (d) remains constant. EXERCISES. The steam point and the ice point of a mercury thermometer are marked as 8 and 2. What will be the temperature in centirade mercury scale when this thermometer reads 32? 2. A constant volume thermometer reisters a pressure of "5 x ' a at the iple point of water and a pressure of 2-5 x a at the normal boilin point. What is the temperature at the normal boilin point? 3. A as thermometer measures the temperature from the variation of pressure of a sample of as. If the pressure measured at the meltin point of lead is 2'2 times the pressure measured at the iple point of water, find the meltin point of lead. 4. The pressure measured by a constant volume as thermometer is 4 ka at the iple point of water. What will be the pressure measured at the boilin point of water ( C)? 5. The pressure of the as in a constant volume as thermometer is 7 ka at the ice point. Find the pressure at the steam point. 6. The pressures of the as in a constant volume as thermometer are 8 cm, 9 cm and cm of mercury at the ice point, the steam point and in a heated wax bath respectively. Find the temperature of the wax bath. 7. In a Callender's compensated constant pressure air thermometer, the volume of the bulb is 8 cc. When the bulb is kept immersec' in a vessel, 2 cc of mercury has to be poured out. Calculate the temperature of the vessel. 8. A platinum resistance thermometer reads when its resistance is 8 Q. and when its resistance is 9 Q. Find the temperature at the platinum scale at which the resistance is 8G Q.

. A concrete slab has a lenth of m on a winter niht when the temperature is C. Find the lenth of the slab on a summer day when the temperature is 35 C.

3 Heat and Temperature 3 9. A resistance thermometer reads R 2' Q, 275 Q, and 5- Q at the ice point ( C), the steam point ( C) and the zinc point (42 C) respectively. Assumin that the resistance varies with temperature as Re - R ( + a6 + p6 2 ), find the values of R, a and p. Here represents the temperature on Celsius scale.. A concrete slab has a lenth of m on a winter niht when the temperature is C. Find the lenth of the slab on a summer day when the temperature is 35 C. The coefficient of linear expansion of concrete is - x " 5 / C.. A mee scale made of steel is calibrated at 2 C to ive correct readin. Find the distance between 5 cm mark and 5 cm mark if the scale is used at C. Coefficient of linear expansion of steel is l'l * 7 C. 2. A railway ack (made of iron) is laid in winter when the averae temperature is 8 C. The ack consists of sections of 2' m placed one after the other. How much ap should be left between two such sections so that there is no compression durin summer when the maximum temperature oes to 48 C? Coefficient of linear expansion of iron x 6 / C. 3. A circular hole of diameter 2 cm is ifiade in an aluminium plate at C. What will be the diameter at C? a for aluminium 23 x ~ 5 / C. 4. Two mee scales, one of steel and the other of aluminium, aree at 2 C. Calculate the ratio aluminium-centimee/steel-centimee at (a) C, (b) 4 C and (c) C. a for steel M x ~>C and for aluminium 23 x ' 5 / C. 5. A mee scale is made up of steel and measures correct lenth at 6 C. What will be the percentae error if this scale is used (a) on a summer day when the temperature is 46 C and 3 ) on a winter day when the temperature is 6 C? Coefficient of linear expansion of steel x " a / C. 6. A mee scale made of steel reads accurately at 2 C. In a sensitive experiment, distances accurate upto '55 mm in m are required. Find the rane of temperature in which the experiment can be performed with this mee scale. Coefficient of linear expansion of steel x " 8 / C. 7. The density of water at C is "998 /cm 3 and at 4 C is /cm Calculate the averae coefficient of volume expansion of water in the temperature rane to 4 C. 8. Find the ratio of the lenths of an iron rod and an aluminium rod for which the difference in the lenths is independent of temperature. Coefficients of linear expansion of iron and aluminium are 2 x / C and 23 x ' / C respectively. 9. A pendulum clock ives correct time at 2 C at a place where 9'8 m/s \ The pendulum consists of a liht steel rod connected to a heavy ball. It is taken to a different place where 9'788 m/s At what temperature will it. ive correct time? Coefficient of linear expansion of steel 2 x V C. 2. An aluminium plate fixed in a horizontal position has a hole of diameter 2' cm. A steel sphere of diameter 2'5 cm rests on this hole. All the lenths refer to a temperature of C. The temperature of the entire system is slowly increased. At what temperature will the ball fall down? Coefficient of linear expansion of aluminium is 23xlO~ 6 / C and that of steel is x " 8 / C. 2. A lass window is to be fit in an aluminium frame. The temperature on the workin day is 4 C and the lass window measures exactly 2 cm x 3 cm. What should be the size of the aluminium frame so that there is no sess on the lass in winter even if the temperature drops to C? Coefficients of linear expansion for lass and aluminium are 9" x " 6 / C and 24 x ~ 8 / C respectively. 22. The volume of a lass vessel is cc at 2 C. What volume of mercury should be poured into it at this temperature so that the volume of the remainin space does not chane with temperature? Coefficients of cubical expansion of mercury and lass are -8 x "V C and 9* x " 8 / C respectively. 23. An aluminium can of cylindrical shape contains 5 cm 3 of water. The area of the inner cross-section of the can is 25 cm 2. All measurements refer to C. Find the rise in the water level if the temperature increases to 8 C. The coefficient of linear expansion of aluminium 23 x ~ / C and the averae coefficient of volume expansion of water 3'2 x 4 / C respectively. 24. A lass vessel measures exactly cm x cm * cm at C. It is filled completely with mercury at this temperature. When the temperature is raised to C, '6 cm 3 of mercury overflows. Calculate the coefficient of volume expansion of mercury. Coefficient of linear expansion of lass 6'5 x 8 / C. 25. The densities of wood and benzene at C are 88 k/m 3 and 9 k/m 3 respectively. The coefficients of volume expansion are *2 x / C for wood " and *5 x 3 / C for benzene. At what temperature will a piece of wood just sink in benzene? 26. A steel rod of lenth m rests on a smooth horizontal base. If it is heated from C to C, what is the lonitudinal sain developed? 27. A steel rod is clamped at its two ends and rests on a fixed horizontal base. The rod is unsained at 2 C. Find the lonitudinal sain developed in the rod if the temperature rises to 5 C. Coefficient of linear expansion of steel '2 x 5 / C. 28. A steel wire of cross-sectional area '5 mm 2 is held between two fixed supports. If the wire is just taut at 2 C, determine the tension when the temperature falls to C. Coefficient of linear expansion of steel is '2 x / C and its Youn's modulus is 2- x " N/m A steel rod is riidly clamped at its two ends. The rod is under zero tension at 2 C. If the temperature rises to C, what force will the rod exert on one of the

$4 Concepts of hysics clamps. Area of cross-section of the rod 2* mm 2. Coefficient of linear expansion of steel 2' * " 6 / C and Youn's modulus of steel - 2' * " Itym \ 3.$

4 4 Concepts of hysics clamps. Area of cross-section of the rod 2* mm 2. Coefficient of linear expansion of steel 2' * " 6 / C and Youn's modulus of steel - 2' * " Itym \ 3. Two steel rods and an aluminium rod of equal lenth lq and equal cross-section are joined riidly at their ends as shown in the fiure below. All the rods are in a state of zero tension at C. Find the lenth of the system when the temperature is raised to. Coefficient of linear expansion of aluminium and steel are aa and as respectively. Youn's modulus of aluminium is YA and of steel is Y.. - Steel Aluminium Steel Fiure 23-E 3. A steel ball initially at a pressure of x 5 a is heated from 2 C to 2 C keepin its volume constant. Find the pressure inside the ball. Coefficient of linear expansion of steel - 2 x / C and bulk modulus of steel -6 x u N/m Show that moment of inertia of a solid body of any shape chanes with temperature as I I (. + 2a), where I is the moment of inertia at C and a is the coefficient of linear expansion of the solid. 33. A torsional pendulum consists of a solid disc connected to a thin wire (a 2'4 x " 5 / C) at its cene. Find the percentae chane in the time period between peak winter (5 C) and peak summer (45 C). 34. A circular disc made of iron is rotated about its axis at a constant velocity co. Calculate the percentae chane in the linear speed of a particle of the rim as the disc is slowly heated from 2 C to 5 C keepin the anular velocity constant. Coefficient of linear expansion of iron -2 x " 5 / C. ANSWERS L (c) 7. (b). (a), (c) 4. (c), (d). 2 C K 3. 6 K ka ka 6. 2 C K (a) 8. (c) OBJECTIVE I 3. (a) 9. (c) 4. (c). (c) OBJECTIVE II 2. all 5. (a) EXERCISES 9. 2 Q, 3-8 x ~ 3 / C, x io ~ 7 / C. -35 m. - cm cm cm 5. (a). (b) 6. (a) 3. (a), (b), (c) 6. (b) 4. (a) (b) -25 (c) (a) -33% (b) - -% 6. 5 C to 25 C x " 4 / C 8. 23: C C cm x 3-8 cm cc cm 24. *8 x ~ 4 / C C 26. zero x ~ N N a YN + 2cu YQ 3. I YS x 8 a x " x " 2 ji.?v

5 . Ice point 2 (L ) L 32 Steam point 8 (L ) L L 32 2 T 2 C L L 8 2 CHATER 23 HEAT AND TEMERATURE EXERCISES a a We know, For constant volume as Thermometer 2.5 T K ressure Measured at M. 2.2 ressure at Triple oint 2.2 T K 6 K a,? T C 373 K, T K T K a, 2? T 273 K, T 2 373K a pa 55 K a T T ice point 8 cm of H steam point 9 cm of H cm 8 t C 7. V T T V V T 273, V 8 CC, V 2 CC 8 T R t 86; R 8; R 9 Rt R 86 8 t R R C 9. R at ice point (R ) 2 R at steam point (R ) 27.5 R at Zinc point (R 42 ) 5 R R (+ + 2 ) R R + R +R 2 R R R K a 23.

6 Heat and Temperature R 42 R (+ + 2 ) R R L?, L m, 5 / C, t 35 L L ( + t) ( ) m. t 2 C, t 2 C, L cm. m, L 2? steel. 5 / C L 2 L ( + steel T).( + 5 ) ( +.).. 2 m. cm 2. L 2 cm, 5 / C tw 8 C ts 48 C Lw L ( + tw) 2 ( + 5 8) m Ls L ( + ts) 2 ( ) m L m.4cm 3. d 2 cm 2 2 t C, t 2 C al / C d 2 d ( + t) 2 2 ( ) m 2.46 cm 4. L st L Al at 2 C Al / C So, st ( st 2) Al ( AI 2) st. 5 / C (a) (b) Al st Al St st Al 4st 4Al ( ( Al st ( ( ( ( Al st 5 2) 2.3 2). AI st 5 5 4) 2.3 4). ) ) (a) Lenth at 6 C L L? T 6 C, T 2 46 C. 5 / C L L L. 5 3 L L % of error % %. 5 3 %.33% L 2 (b) T 2 6 C L L % of error % %. 5.% L L

7 6. T 2 C, L.55mm.55 3 m t 2? st 6 / C We know, L L T In our case, I 6 (T +T 2 ) ± 3 T 2 T C or C The expt. Can be performed from 5 to 25 C 7. ƒ C.98 /m 3, ƒ 4 C /m 3 ƒ ƒ C 4 C T As density decreases Iron rod Aluminium rod L Fe L Al Heat and Temperature Fe 2 8 / C Al 23 8 / C Since the difference in lenth is independent of temp. Hence the different always remains constant. L Fe L Fe ( + Fe T) () L Al L Al ( + Al T) (2) L Fe L Al L Fe L Al + L Fe Fe T L Al Al T LFe Al L : 2 2 Al Fe m/s 2, m/s 2 l l2 T 2 T Steel 2 6 / C T 2 C T 2? T T 2 l 2 2 l ( T) l l ( 2 l( T) 2 T T T T T T C 2. Given d St 2.5 cm, d Al 2. cm S 6 / C Al 23 6 / C ds 2.5 (+ s T) (where T is chane in temp.) ds T d Al 2(+ Al T) T The two will slip i.e the steel ball with fall when both the diameters become equal. So, T T ( ) -6 T.5 T T) Steel Aluminium

8 Now T T 2 T T 2 C [ T C iven] T 2 T + T The final lenth of aluminium should be equal to final lenth of lass. Let the initial lenth o faluminium l l( Al T) 2( ) l( ) 2 ( 9 6 4) l(.96) 2 (.36) l 2.2 cm.9994 Let initial breadth of aluminium b b( Al T) 3( ) b 3 ( 9 ( ) 4) V CC, T 2 C V H? H.8 4 / C 9 6 / C T remains constant Volume of remainin space V V H Now V V ( + T) () V H V H ( + H T) (2) Subactin (2) from () V V H V V H + V T V H H T V 4 H.8 VH V 6 H 9 V HG CC. 3.8 cm Heat and Temperature Volume of water 5cm 3 Area of cross section of can 25 m 2 Final Volume of water 5( + ) 5[ (8 )] 5.2 cm 3 The aluminium vessel expands in its lenth only so area expansion of base cab be nelected. Increase in volume of water.2 cm 3 Considerin a cylinder of volume.2 cm 3. 2 Heiht of water increased cm 24. V CC T C, V HG V.6 cm / C, H?, / C V H v HG ( + H T) () V v ( + T) (2) V H V V H V + V H H T V T.6 H H / C 25. ƒ 88 K/m 3, ƒ b 9 K/m 3 T C,.2 3 / C, b.5 3 / C The sphere beins t sink when, (m) sphere displaced water

9 Vƒ Vƒ b ƒ ƒb b () () ( ) () 2 (32 8) 3 () C 83 C 26. L C A lonitudinal sain develops if and only if, there is an opposition to the expansion. Since there is no opposition in this case, hence the lonitudinal stain here Zero C, 2 5 C steel.2 5 / C nitudinal stain? L L Stain L L.2 5 (5 2) A.5mm m 2 T 2 C, T 2 C s.2 5 / C, Y 2 2 N/m 2 Decrease in lenth due to compression L Sess F L FL Y L Sain A L AY Tension is developed due to () & (2) Equatin them, FL L AY F AY (2) () 23.Heat and Temperature m.2-5 (2 ) N C, 2 C A 2mm m 2 steel 2 6 / C, Y steel 2 N/m 2 Force exerted on the clamps? F A Y L YLA Y F L YA Sain L L N 3. Let the final lenth of the system at system of temp. C l Initial lenth of the system l When temp. chanes by. Sain of the system total sess of system But the total sain of the system total youn' s mod ulusof of system Now, total sess Sess due to two steel rod + Sess due to Aluminium s s + s ds + al at 2% s + 2 Al Now youn modulus of system s + s + al 2 s + al Steel Aluminium Steel 23.5

10 2ss s Sain of system 2 2ss s 2 alal 2 l l al 2s s al s al s s al al Heat and Temperature 3. The ball ies to expand its volume. But it is kept in the same volume. So it is kept at a constant volume. So the sess arises V v B V B V B B (2 2) pa. 32. Given Moment of Inertia at C Coefficient of linear expansion To prove, ( + 2) Let the temp. chane to from C T Let R be the radius of Gyration, Now, R R ( + ), MR 2 where M is the mass. Now, MR 2 MR 2 ( + ) 2 MR 2 ( + 2) [By binomial expansion or nelectin 2 2 which iven a very small value.] So, ( + 2) (proved) 33. Let the initial m.. at C be T 2 K ( + 2) At 5 C, At 45 C, T 2 T 2 2 T 2 9 T (from above question) ( 2) 2 K ( 245) 2 K ( 25) 2 K ( 9) K T2 % chane T.959% % 34. T 2 C, T 2 5 C, T 3 C.2 5 / C remains constant V V (I) (II) R R Now, R R( + ) R + R R From (I) and (II) V V V R R.36R V.36 V (.36V V) % chane V ( ) K

Class 11 Physics NCERT Exemplar Solutions Thermal Properties of Matter

Class 11 Physics NCERT Exemplar Solutions Thermal Properties of Matter Q1. A bimetallic strip is made of aluminium and steel (a AI > astee ) On heating, the strip will (a) remain straight (bj get twisted