JEE [MAIN + ADVANCED] FLUID MECHANICS eel ee) Weg Cun ea) BANSAL CLASSES - PRIVATE LIMITED Ideal for ScholarsFLUID_MECHANICS Ra i) ‘© For the system shown in the figure, the cylinder on the leftat L has a ‘mass of 600kgand a cross sectional area of 800 em?. The piston on the right, aS, has cross sectional area 25cm? and negligible weight. If the apparatus is filled with oil.(p= 0.75 g/cm?) Find the force F 8 required to hold the system in equilibrium. aaah: spherical tank of 1.2m radius is half filled with oil of relative density 0.8 . Ifthe tank is given a horizontal acceleration of 10nvs?. Galculate the inclination of the oil surface to horizontal and maximum guage pressure on the tank: ‘The volume of an air bubble is doubled as it rises from the bottom of a lake toits surface. If the atmospheric pressure is H m ofmercury & the density of mercury is n times that of-lake water. Find the depth ofthe lake. «92 Aveta unin the penta ends cotins new Water is poured in one limb until the level of mercury is depressed 2cm in that limb, Whats the length of water colurmn when this happens, ‘Anopen cubical tank completely filled with water is kept ona horizontal surface. Its acceleration is then slowly increased to 2mv/s* as shown inthe Fig. The'side 1m of the tank is Im. Find the mass of water that would spill out ofthe tank. bean? im Qn ‘iran object weighs ISN, when immersed completely in water the same object weighs 12N. When 7 immersed in another liquid completely, it weighs 13N. Find specific gravity of the object and the specific gravity of the other liquid. A solid ball of density half that of water falls freely uncer gravity from aheight of 19.6 mand then enter water. Upto what depth wil the ball go? How much time will t take to come again tothe water one 2 Neglect ar resistance & velocity effees in Walee el y pean: In asonometer wire the tension is maintained by suspending a 50.7 kg mass from the free’end of the wire. The suspended mass has a volume of 0.0075 m?, The fundamental frequency of the wire is 260 Hz. Find the new fundamental frequency ifthe suspended mass is completely submerged in water. Ge heya cae =n dency 5 floats vertically in aliquid a a7 — 4 f density p.as shown in Fig (a). v of . (o Slowtat perish pulled slightly up& released & Al ol find itstime period, Neglect change in liquid level. ® » Find the time taken by the rod to completely immerse when released from position shown n(b), Assume that it remains vertical throughout its motion. (tak® g= 72 m/s*) . ‘A wooden stick of length and radius R and density p has asmall metal piece of mass m (of riegligible volume) attached to its one end. Find the minimum value for the mass m (in terms of given parameters) that would make the stick float vertically in equilibrium ina liquid of density o (>p).FLUID MECHANICS ‘A uniform solid cylinder of density 0.8 gm/em flotsin equi of two non mixing liquids A and b withits axis vertical. The densities ofthe liquids Aand | s are 0.7 gm/em® and 1.2 glom’, respectively. The height of liquid Aish,=1.2cm.The | length ofthe part ofthe cylinder immersed in iquid Bish, = 0.8 em. 3 ef” Find the toa fore exerted by liquid A onthe eylinder. Find the length of the part ofthe cylinder in air. (c) _‘Thecylinder is depressed in such a way thats top surface is just below the upper surface ofiquid A and “ is then released. Find the acceleration of the cylinder immediately after it is released. ‘A thin rod of length L & area of cross-sectton S is pivoted at its lowest point P inside a oO stationary, homogeneous & non-viscous liquid (Figure). The rod is free to rotate ina vertical plane about a horizontal axis passing through P. The density d, of the material ofthe rod is smaller than the entity d, ofthe liquid. The rodis displaced by asmall angle @ from its equilibrium position and then released. Show that the motion of the rod is simple harmonic and determine its angular frequency in terms of the given parameters. Q.12 Place a glass beaker, partially filled with water, in a sink. The beaker has a mass 390 gm and an interior volume of $00em?. You now start to fill the sink with water and you find, by experiment, that if the beakers less than half full, it wll float; but if tis more than half full, itremains on the bottom ofthe sink as the waier rises to its rim. Whatis the density of the material of which the beaker is made? wy ‘Atest ube of thin walls has some lead shots in it at its bottom and the system floats vertically in water, sinking by a length ,= 10cm. A liquid of density less than that of water, is poured into the tube till the levels inside and outside the tube are even. If the tube now sinks to a length 1=40em, the specific gravity ofthe liquid is : © ot ‘Avertical cylindrical container of base area A and upper cross-section ose area A, making an angle 30° withthe horizontal is placed inan openrainy - inal field as shown near another cylindrical container having same base area , , Find the ratio. ‘of rates of collection of water in the two containers. m ©. os Water is pumped from a depth of 10 m and delivered through a pipe of cross section 10 mi? upto a heightof 10 m, Ifitis needed to deliver a volume 0.2 m? per second, find the power regained. [Use g= 10 m/s" ow ‘A laminar stream is flowing vertically down from a tap of cross-section area 1 cm’, Ata distance 10cm below the tap, the cross-section area of the stream has reduced to 1/2.cm, Find the volumetric flow rate f water from the tap. 0-9 main iti pene side walls at heights of h, and h, respectively such thatthe range of efflux at the bottom of the vessel is same. Find the height off hole, for which the range of efflux would be maximum, & A large tankis filled with two liquids of specific gravities 2c and c. Two holes are spec 77 madeon he wall ofthe tank as shown, Find the ratio ofthe distances fom O ofthe points on the ground where the jets from holes A& B strike. “atthe rate 100 cm’s"!. Find the height of water in the vessel under steady state, a open top container of negligible mass & 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 aliquid of density pandmass m, . Assuming thatthe liquid stats lowing outhorizontally through the hole at t= 0, calculate G _, the acceleration of the container and a7 its yelocity when 75 % of the liquid has drained out. Calculate the rate of flow of glycerine of density 1.25 x 10° kg/m’ through the conical section of a pipe if the radii of its ends are 0.1m & 0.04m and the pressure drop across its length is 10N/m?, 9% Anonviscous liquid of constant density 1000 kg/m? flows na streamline motion along a tube of variable eross section. The tube is kept inclined inthe vertical plane as showm in the figure, The area of cross section ofthe tube at two points P and Q atheights of 2 meters and 5 meters are respectively 4 10-'m? and 8 10° m’. “The velocity of the liquid at point Pis | m/s. Find the work done per unit volume by the pressure and the gravity forces as the fluid flows from point P to Q. 8 ak Asiphon has ‘uniform circular base of diameter “-= ‘cm withits crest vel A1.8mabove water level asin figure, Find oe locity of flow harge rate ofthe flow in m/sec. OO sete tetova, [Use Py = 105 Nim? & g= 10m/s"] (G24 Aietof aterhaving velocity= 10 m/sand.sream cros-ection=2 mn? his fat plate perpendicularly, with the water splashing out parallel to plate. Find the force that the plate experiences. 028 ‘A U-tube open at both endsis partially filed with water (figure-(a). Oil having a density of 750 kg/m? is then poured into the right arm and forms a column L = 10 cm in height (figure-(b)). The right arm is shielded from any air motion while airis blown across the top of the left arm until the surfaces ofthe twa liquids are atthe same height (figure-(c)). Determine the speed of the air (in m/s) being blown across the leftarm. (Take the density of air as 1.25 kg/m?.) 4. v Siijeld @ © Page #23FLUID MECHANICS jus R and thickness d (<> db ‘Two soap bubbles with radii rand (r,> r,) come in contact. Theit common surface has a radius of curvature r. Find values of, cand B. ‘A.cube with a mass ‘m* completely wettable by water floats on the surface of water. Each side of the cube is ‘a’. What isthe distance h } between the lower face of cube and the surface of the water if surface Ch \. tension is 8. Take density of water as p,, Take angle of contact as zero. «68 in on re eter pe ps aes spared isenesh=0.10mm, ae <- awater drop of mass m=70 mg was introduced between them. The wetting is assumed to be complete. ‘Twoamns ofa U-tube have unequal diameters d, = 1.0 mm and d, = 1.0 cm. If water (surface tension 7x 10? N/m) is poured into the tube held in the vertical position, find the difference of level of water in the U-tube. Assume the angle of contact to be zero. 31 Water is pumped into a horizontal capillary tube with an internal diameter of d= 2 mm so that a column ‘h= 10cm long is formed. How many milligrams of the water will flow out of the tube if it is placed vertically ? Consider wetting to be complete. S= 0,075 Nim. (Take x= 22/7). ow ‘A bubble having surface tension T and radius R is formed on a ring of radius b (b << R). Airis blown inside the tube with velocity v as shown. The air molecule collides perpendicularly with the wall ofthe bubble and stops. Calculate the radius at which the bubble separates from thering. @ Soe = of ‘A spherical ball of radius 1 « 10~ mand density 10* kg/m? falls freely under gravity through a distance h before entering a tank of water. If after entering the water the velocity of the ball does, not change, find h. The viscosity of water is 9.8 < 10-°N-s/m?, (34 A spherical ball of density p and radius 0.003m is dropped info atube containing a viscous fluid filled up to the 0 em mark as shown in the figure. Viscosity of the fluid = 1.260 Num and its density p,= p/2 = 1260 kgm. Assume the ball reaches. terminal speed by the 10 em mark. Find the time taken by the ball 10 traverse the distance between the 10cm and 20cm mark, (g= acceleration due to gravity = 10 ms)[SINGLE CORRECT CHOICE TYPE] 9 ‘Anopen cubical tank was initially fully filled with water. When the tank was accelerated on a horizontal ol ‘The area of cross-section of the wider tube shown in figure is wt plane along one ofits side it was found that one third of volume of water spilled out, The acceleration was (Aye B)293 (©3g2 (D) None 800 cm?. Ifa mass of 12 kgis placed on the massless piston, the difference in heights hin the level of water in the two tubes is : (A) 10cm (8)6cm (C)1Sem ()2cm Some liquid is filed in a cylindrical vessel of radius R. Let F, be the force applied by the liquid on the bottom of the cylinder. Now the same liquid is poured into a vessel of uniform square cross-section of side R. Let F, be the force applied by the liquid on the bottom of this new vessel. (Neglect atmosphere | ressure) Then woh oe E (A)F, = nF, @)F,-2 OF, = VaF, ()F, =F, liquid of mass | kg is filled inva flask as shown in figure. The force exerted by the flask on the liquid is (g= 10 m/s2) [Neglect atmospheric pressure}: (A) 10N B)greaterthan 10N (C)lessthan 10N — (D) zero Anopen-ended U-tube of uniform cross-sectional area contains water (density 1.0 gramy/centimeter®) standing initially 20 centimeters from the bottom in each Sem arm, An immiscible liquid of density 4.0 grams/ centimeter? is added to one arm # untila layer 5 centimeters high forms, as shown inthe figure above. Whatisthe | $f ts ratio h,/h, of the heights of the liquid in the two arms? 4 3 B)52 © (p32 ‘The pressure at the bottom of a tank of water is 3P where P is the atmospheric pressure. Ifthe water is drawn outtill the level of water is lowered by one fifth, the pressure at the bottom of the tank will nowbe# (A)2P (B) (13/5)P (C)(8/5)P ()@/5)P ‘A.U—tube having horizontal arm of length 20 ci, has uniform cross-sectional arca= lem? Itis filled with water of volume 60 cc. What volume ofa liquid of density 4 g/cc should be poured from one side into the U-tube so that no water is leftin the horizontal arm ofthe tube? (A)60cc | * (B)45 ce (©) 50ce (D)35 ce A bucket contains water filled upto a height = 15 em. The bucket is tied to a rope which is passed over a frictionless light pulley and thé other end of the rope is ted to a weight of mass which is halfof that of| the (bucket + water). The water pressure above atmosphere pressure at the bottom is (A)0.5 kPa (B) 1 kPa » (©)SkPa_ (D) None of theseFLUID MECHANICS having density p. The gauge pressure atthe centre of the cubical vessel is L L A778 ®B) zP(eta) L L (©) Z0a @) 7 Ae-a) 946 Aliph semi cylindrical gat ofradiusR is pjovted atts midpoint O ofthe diameter as shown inthe figure holding liquid of density p. The force F required to prevent the rotation ofthe gate is equal to (A) 2nRpg (B)2pgR% 4) 3 x oO 2s (D) none of these i Lot ‘Asolid metallic sphere of radius is allowed to fall freely through air. Ifthe frictional resistance due to air is proportional to the cross-sectional area and to the square of the velocity, then the terminal velocity of the sphere is proportional to which of the following? Ar @®)r © Myre Q.19/ Acontainer of large surface area is filled with liquid of density p. Acubical block ofside edge aand mass Mis floating in it with four-ffth ofits volume submerged. IFa.coin of mass m is placed gently on the top surface of the block is just submerged. Mis (A)4m/S (B) m/s (4m (D) sm us A boy carries. fish in one hand and a bucket (not full) of water in the other hand . Ihe places the fish in the bucket, the weight now carried by him (assume that water does not spill) : (A) isless than before (B) ismore than before” (C) is the same as before (D) depends upon his speed QA Acorkof density 0.5gem-? floats on acalm swimming pool. The fraction ofthe cork’s volume which is under water is (a)o% B25% (10% (0) 50% Q.197 A small ball ofelative density 0.8 falls into water from aheight of 2m. The depth to which the ball will sink is (neglect viscous forces): (a)3m @)2m (6m 4m A small wooden ball of density p is immersed in water of density o to depth h and then released. The height H above the surface of water up to which the ball will jump out of water is. th wt ® (s-}s Ob @)20FLUID MECHANICS Abhollow sphere of mass M and ‘immersed in a tank of water (density p, float if it were set free, The sphere: ‘to the bottom of the tank by two with the horizontal as shown in the figure. The tension T, in the wire is : 4 AR} BFR Pas—-Mg yoo @) F#R’oyg-Me v2 3 4 aR’, g-Mg 4 © ap ©) Z7R’pwe+Mg identical cylinders haye # hole of raditis a (a<F, (OF >F, ()F,=F,40 we A sphere ofradius R and made of material of relative density o has a concentric cavity of radius r. Itjust floats when placed ina tank full of water. The value of the ratio R/r will be wl)” a(t=)" o(t2)” oz)” 9207 A body having volume V and density pisatached the bottom ofa container as shown. Density of the liquid is d(>p). Container has a constant upward acceleration a. Tension inthe stringis (A)VIDg-p(gta)]_ B) Vigta)(d-p) (©V-p)g (@) none a" The frequency.of a sonometer wire is f, but when the weights producing the tensions are completely immersed in water the frequency becomes f/2 and on immersing the weights in a certain liquid the frequency becomes 73. The specific gravity ofthe liquid is: 4 16 15 32, “> es OF OF QP Ketindica bléck of area of cross-section A and of material of density pis placed ina liquid of density one-third of density of block. The block compresses spring and compression in the spring is one-third of the length of the block. If acceleration due to gravity is g, the spring constant of the spring is: (A)pAg (B)2pAg (©)2pag/3 (@) pAg/3