Mechanical Properties of Solids & Fluids

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Mechanical Propertieso f Solids 8 luids1 i EE dvancedl IIT JEE
1. A wire of length L and cross sectional area is made of amaterial ofyou ng s mod ulus Y. If the wire is stretched by.an.................mountx, theworkdone is 1987 2 Marks)2 A solid sphere ofradius ~ l n a d ef a ntatlrial of bulk modulusK is sunou nde d by a liquid in a cylindrical container. A
massless pistion of ar eaA floats on the surface of the liquid.When a mass M is placed on the piston to compress, . the liquid the fractional change in the radius of the sphere,6 RIR is ............ 1988 2 Mark).
3 . . A piece of metal floats on mercury. The coefidients of.volume expansion of the metal and mercu ry are y, a n d y2respectively. If the temperatures of both mercury and themetal are.increased by an amount A T , the fraction of theVolume of the metal submerged in mercury changes by thefactor ..... .... ...... 1991  2 Mark) 4.
4. A horizon tal pipeline c ar he s wate; in a streamline flow.At a point along the pipe, where the cross sectionalarea is 10 cm2, the water velo city is 1 ms and th:pressure is 2000 Pa. The pressure ofwater at another
Water in a closed tube (see Fig) is heated with one rmvertically placed above a lamp. Wafer will begin to circulatealong the tube in counterclockwise direction.
1983  2 Marks)
A block o f ice with a lead sho t embedded in it is floating onwater contained in a v essel. Th e temperature of the systemis maintained at 0C a s the ic e melts. W hen the ice meltscompletely the level of water in the vessel ris es .
1986 3 Marks)point where the crosssectional area is 5 cm2, is... Pa.(Density of water = l o 3kg.m ) 1994 2 Marks) [@ e am1 A man is sitting in a boa t which is floating in a pond. If th em an driik s some water from the pond, the level ofthe water
in the pond decreases, 1980)2 A barometer made of a ve ry nm ow ube (seeFig). is placed
at normal temperature and pressure. The coefficientofvolume expansion of mercury is 0.00018 per Coand that ofthe tube is negligible. The temper ature of mercury in thebarometer is now raised by l0C hut the temperature of theatmosphere does not change. Then the mercury height intheiube remains unchanged: 1983 Marks)
1. A vessel containing water is given aconstant accelerationa towards the right along a straight horizontal path. Which
of the following diagrams in Fig. represents the surface ofthe liquid ? 1981 2 Marks)
(a) , b (c) .2 The following four wires are made o f the same material.
which of these will have the largest extension when thesame tension is applied ? 1981 2 Marks)
, . (a) length=50 cm ?diameter= 0.5 mm(b) length= 100 cm, diameter = I(c) length= 200 cm, diameter = mm
. d) Iength=300cm, diameter=3k.
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P 146 Topicwise Solved paperb PH YS ICS. .3 A [email protected] of uniform c ross section (see Fig) is partially filledwith a liquid I. ~ " g t h e r iquid I1 which doe s not mix.with
liquid I is poured into on e side. It is found that, the liquid .levels of the two sides o f the tube a re the same, while thelevel of liquid I has risen by 2 cm. If the specific gravity of . ,liquid I is 1.1, the specific grav ity of liquid I must he
. I983 1 Mark
. .(a)' ~g (b) Mg Vpg(c) M g + lrR2hpg (d) pg(V+?R2h)
7 A wooden block, with a coin placed on its top, floats inwa ter as shown in figure.. The distance e and h are shown(a) 1.12. (b) 1.1 . . here. After som e time the coin falls into the water. Then
(cj . 1.05 (d) 1.0 . . . 2002s)4 A homogeneous solid cylinder of length L (L

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MECHANICAL PROPERTIESOFSOLI S FLUIDS11. A glass tube of uniform intemal radius (r) has a valveseparating the two identical ends. Initially, the valve is in a . .tightly closed position. End 1 has a hemispherical soapbubble of radius r.: ~n d h is subhemispherical soap bubble
as shown in figure. Just after opening the valve, (2008)
(a) air from end 1 flows towards end 2. No change in the . .volume of the soap bubbles(b) air,fromend 1 flows towards end 2. Volume ofthe soap 3bubble at end 1 decreases . . ~
. (c) no changes occurs. . (d) . air from eGd 2 flows towards end 1. volum e of the soap .
bubble at end 1 increases ~ .12 A thin uniform cylmdrical shell, closed at both knds, ispartially filled withwater. It is flo atingv ert~c allyn water inhalfsubmerged state: If p, is the relative density of the 4, . material of the she li with respect to water, then the correctstatement is that.the shell is (20 12ll)\ (a) more than halff illed if p,is less than 0.5.
b) more than halffilled if o is more than 1O
(a) the balance A will readmo re than 2 kg(b) the balance B will r iad p or e tha n5 kg . .(c) the balance A will read less than 2 kg and will readmore than 5 kg . .: d) th e balance A andBw illread2 kg arid SkgrespectivelyA vessel containsoil (density = 0.8 gm/cm3) over mercury(den sity= 13.6 gm cm3).Ahomogeneous sphere floats withhalfits volume immersed in mercury and the other haifin oil.The density of th6 material of the sphere in gm/cm3 s
.(198 8 2 Mark). .(a) 3 3 . . @ ) 6.4 . , .(c) 7 2 . . (d) 12.8Two rods 'of different materials ha ~i n~ co ef ic ie nt sfthermalexpansion al 2 and Young's m odulii Y1,Y2 respectivelyare fixed between two rigid massive walls. The rods areheated such that they undergo the same increase in,\ .(c) halffilled if p is morei ian 05 . . . . temperature.T here is no bending of the rods. If 41: ax =(d) less than halffilled if p, is less than 0.5. . 2 : 3, the thermal stresses d&elop& iii the two rods are13. One end of a horizo'ntal thick copper wire of length 2L au dradius 2R is welded to an end of another horizontal thin' equal provided Y1 Y2 is equal to . (1989 2 Mark)
. copper wire of length L and radius R. when the arrangementis stretched by applying forces at tw ends, the ratio of the . ,elongation in the thin wire to that in the thick wire is(JEE Adv. 2013) 5(a) 025 (b) 0.50(c) 2.00 . . (d) 4.00
K 8 8 8   1 1 . ~ 4 61 A body floats in a liquid contained in a beaker. The wholesystem as shown in Figure falls freely under gravity. Theupthrust on the Body is (1982 3 Marks)
(a) zero . . .(b) equal fo the weigh t of the liquid displaced .'
. (c) egu alto the weight of the body in air(d) equal to the weight of the immersed portion o f the body
2 The spring balance A reads 2 kg with a block m suspendedf r om it. A balance Bre ads 5 kg when a beaker with liquid isput on the pan of the balance. The two balan ces are now s oarranged that the'hanging mass is inside the liquid inthe.beaker as shown in the figure. In this situation:. . (1985 2 Marks)
(b) 1 : la) 2 : 3(c) 3 : 2 (d) 4 : 9Water from tap emerges vertically downwards with aninitial spped of 1 Om sI. The crosssectional area of the tapis 10' m2. Assum e that the pressure ii constant throughou tthe stream of water, and that the flow is steady. The cro sssectional area of the stream 0.15 m beiow thctap is. ~ (199 8s 2 Marks)
(b) 1 . 0 ~0 m2a) 5 . 0 ~O4m2(c) 5 . 0 ~ 1 O  ~ m ~ (d) 2 . 0 ~0 m2A solid sphere of radius R and density p isattached to oneend of a m asslessspring of force cdnstant k The other enda f the spring is codnected to another solid sphere of radiusR and d e n ~ i t y . 3 ~ .hec om plete arrangement is placed in aliquid of den sity 2p an d is allowed to reach equilibrium. The.'correct statemen tcsj is (are), (JEE Adv. 2013)
, . . ,4 n ~ ~ p g'(a) The net elongafion of the spring is k. .
(b) The net elongation of the spring is3 k \\,\ .,
(c) The light sphere is partially subm erged . \(dj The light sphere is completely submerged , , .. .,,
\
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P 148 Topicwise Solved Pa per s PHY SICSI@ 9. A container of large uniform crosssectional area A restinge : on a horizontal surface, holds hvo immiscible, nonviscous
A column of'm ercu~ y.of 10 cm length is contained in themiddle of anarrow horizontal 1ml ong tuh e which is closedat both the ends. Both the h alves o f the tube contain air ata pressure of 7 cm o f mercury. By.what distance will thecolumn of mercury be displaced if the tube is held vertically?1978)A point mass m is suspended at the end of a massless wireof length 1and cross section A. If Y is the Young's modu lusfor the wire; obtain the frequency of oscillation forthe simpleharinonic motionalong the vertical line. 1978)A cube of wood supporting 200 gm mass just floats in water.When the mass is removed, the cube ruses b y 2cm. W hat isthe size ofth e cube? ~ ~ 1978)A beaker containixig.watei is placed on the pan of balancewhich shows areading of M gms. Al um pof sugar of mass mgm i and volume Vcc. is now suspended by a thread in sucha way that it is completely immersed in water withouttouching the beaker and without atiy overflow of water.What will be the reading of the balance just when the lumpof sugar is immersed? How will the rea ding change as thetime passes on? 1978)A boat floating in a w ater tank is carrying a num ber of largestones. If the stones are unloaded into water, what willhappen to the water level? 1979)'Two identical cylindrical vesse ls with their bas esa t the samelevel each contain a liquid o f density p (rho). Th e height ofthe liquid.in one vessel is h , and in other is h,. The area ofei ther baseis A . What is the work done by gravity inequalizing the levels when the two vessels are connected ?1981 4 Marks)A woodenplank of length 1 m and unifoq crosssec tion ishinged at one end to the bottom o f a tank as shown in figThe tankis filled w ith.water upto ah eig ht 0.5 m. The specificgravity o f he plank is 0.5. Find the angle 9 that the plankmakes with the vertical in the equilibrium position. (Excludethe case 0 =On) 1984 8 Marks)
I
A ball of densiw di s dropped on to a horizontalsolid surface.It bounces elastica11~rom the surface and returnsto itsoriginal position in aSime t , .Next, the ball is released anditfa113 through the sa me height befor e striking the surface ofa liquid of d ensity o f d,, I992 8 Marks)(a) If d < d,, obtain an expression (in te r~ ns f d, t , and d,for the time tzth e ball takes to come back to the positionfrom which it was released. .b) '1s the motion o f the ha ll sim ile h&rm onic?
(c) I f d = d,>, how does the speed of the ball depend on itsdepth inside the liq uid? Neglect a ll frictional and otherdissipative forces. Ass um eth e depth of the 1 iq ui d. t~be large.
and incompressible liquids of densities d and 2 4 each ofheight HI2 a s shown in the figure. The lower density liquidis open to the atmosphere having pressure P
t X  +(a) A homo geneous solid cylinder of: length L(L

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. .MECH NIC LPROPERTIES O F SOLIDS FLUIDS P14912. A wooden.stick of length L, radius R and density p has a 16. A tube has two area o f crosssections as shown in figure.small metal piece ofmass m (ofnegligible volunie) attached to The diam etersofthe tube are 8 mm an d2 mm. ind range ofits one end. Find the,minimum value fo r the mass n (in terms water falling on ho rizontal surface, if piston is moving withof given parameters) that would mpke the stick float vertically a constant velocity of 0..25 d s , h = 1.25 m g= 1 0 d s 2 )in equilibrium in liquid of density G > p).(1999  I0Marks) .. = 2004  2 Marks)~a==.L ; ..... .13. A uniform solid cylinder of density 0.8 g/cm3 floats in :~~. izxa; =F==.... . .equilibrium in a com bination of two nonmixing liquids A . ~==. .= .L= . . ..and B with its axis vertical: The densities of the liquids A . . ~

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Situation I :The water is remo ved fro m

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MECHANICALPROPERTIES OF SOLIDS FLUIDS P151I J Main/ AlEEE. 1. A spring of force donstant 80 0 Nlm has an e*tension of 5cm. The work done in extend ing it from 5 cm to 15 cm is VgP@ ~.(a) 165 (b) 8J [ZOO21 ~.
. (c) 32 J (d) 2 4 J2. A wire fixed at the up,per end stretch es by length by ( 4 vgip, Pz)applying a force F. The work do ne in stretching is [2004] k(a) 2Ft (b) Ff. 10. 'A j a r is filled with two nonmixing liquids 1 and 2 havingF . F t densities p , and, p, re spectively. A solid ball, made of a(c) z ( 4 mateiial of density p, , s dropp ed in the jar. It comes to
3 Spherical balls of radius R are falling in a viscou s fluid of equilibrium in the position shown in the figure.Which ofviscosity 11 with aveloc ity v . The retarding viscous force the following is true for p,, plan d p,?, [ZOOS/acting on the spherical ball is [ZOO41(a) inversely proportional to both radius 'R' and velocity ............, ............................. b) directly proportional to both rad ius Z? andvelocity 'v'
. (c) ,directly proportional to R but inversely proportionalto v ............... . (d) inversely proportional to R but directly proportional'to velocity 'v'4. If two soap bubbles of different radii are connected hy a (a) P ~ < P IPz @ Pl>P,>Pztu be . /Z 004/, ( c ) . P < P z < P ~ . (d) P,

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......................................................... ..........................
..........................................water.............
..................:::::: oil :::::::. . . . . . . . . . . . .........................................................................
P152 TopicwiseSolved Pape rs PH YSIC S12. T ~ Oires are made o f the same material and have the same 18. TWO mercury drops (each ofmdi"s r ) merge to form biggervolume. However wirk 1 has cso ss~sec tional reaA and wire drop. The surface energy of the bigger drop, if T is the . .2 has crosssectional arqa 3A. If th e le'ngth of wire 1 increases ,. surfac e tension, is : [ZOllRS]by x on applying force F how much force is needed tostretch wire 2 by the sam e amount? . . [2009/ . (a) 4nr2T (b) 2nr2T(a) 4 F (b) 6 F (c) 28 3rrr2T (d) 2'1 3rr r2 ~6 ) 9 F (d) F1 3; A b a ll i s m a d e o f a m a t e r ia l o f d e n s i t y p w h e r e 19. If a ball of steel (density p = 7.8 g cmJ) attainsa terminalvelocity of 10 cm S I when falling in water (Coefficient ofPail < P < Pwater w ith poi and p representing the viscosity q,,,, = 8.5 x lo 4 Pa$, then, its terminal velocity .densities of oil and wa ter, respectively. The oil and w ater are . .immiscible. Ifth e above b all is in equilibrium in a mixture of 'in glycerine (p = 1.2 g = 13.2 P a s) would be, nearlythis oil and water, which o ft he following pictures represents [2011RS]its equ ilibrium position ? ~ ]2010 ] (a) 6.2 5 x 10" cm SI b) 6.45 x cm sI(c) 1.5 1o ~m sI (d) 1.6 ~1 0" cm s '20. If a spring of stiffness k is cut into parts 'A' and 'B '.of. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . , length P :t 2 :3, then the,stiffness of spring A is... ................. . . . . . . . . . . . given by. . . . . . . . . . . . [ZOllRS]. . . . . . . . . . . .. . . . . . . . . . . . 3k 2k. . . . . . . . . .
(a)(a)  b)
b) 5............................water (c) k 5.............. , ( d ) . % '..............21. A thin liquid film formed between a Ushaped wire and a lightslider supports a weight of 1. 5x N (see figure). The, . . length of the slider is 30 cm and its weight negligible. The
. .
(c). .14. Two identical charged spheres are suspended by strings of . .equal lengths. The strings ma ke an angle of 30' with eachother. When suspended in a l iquid of density 0.8g ~ m  ~ ,he
. angle remains the same. If density of the material of the . W .sphere is 1.6 g c m 3 th e dielectric constant o fth e liquid is[zorol (a) 0.0125 NmI (b) 0.1 NmI(a) 4 b) 3 (c) o . o ~ N ~ I . . (d) 0.025 ~m ( 4 2 (d) 1 . . 22. A uniform cylinder of length L and mass M having cross.1 5. Work done in increasing the size of a soap bubble from a sectional are a A is suspended, w ith its length vertical, from . .radius of 3 cm to 5 cm is ne,arly (Surface tension o f soap a fixed point by a massless spring such that it is halfsolution= 0.03 NmI) [2Oll] submerged in a liquid of density t equilibrium position.(a) 0.2nmJ b) 2nmJ The extension xo of th e spring when it is inequilibrium is:(c) O.4nmJ (d) 4nm .J, ~~
[ JE E Main2013 ]16. Water is flowing continuously fiom a tap having an internaldiameter 8 x m. The water velocity a s it leaves the tap is0.4 msI. The dia me tero f the water stream at a distance.2 x 10I m b elow the tap is close to: [ZOll](a) 7.5 x 104m b) 9.6 x104m .(c) 3.6 X 10J m (d) 5.0 x 10 m17. AmetalrodofYoung's modulus Yhdcoeficientofthermal 23. Assume that a drop of liquid evaporates by decrease in itsexpansion a is held a t its two ends such that its len gth . sur f ace energy , so tha t i t s t empera ture r emainsremains invariant..lf its temperature is raised by PC , the unchanged.What should be the minimum radius of the drop1inear.stress developed in it is [2011RS] for this to be possible? T he surface tension is T, density of
. . .. liquid is p and L is its latent heat of vaporization.(a) . (t b) ~ t . [JEEMain2013] .1 t( 4 ~. , (d) (a) P V ~ , b) JT7pL(c) TIPL (d) 2TIpL
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