SECTION (C) : RADIUS OF CURVATURE AND DYNAMICS OF CIRCULAR MOTION

Transcription

1 SECTION : KINEMATICS OF CICULA MOTION A. Figue shows a cicula path taken by a paticl If the instantaneous velocity of the paticle is v = ( /s) i ( /s) j. Though which quadant is the paticle ovi when it is tavelli (a) clockwise and (b) unte clockwise aound the cicle? SECTION : ADIAL AND TANGENTIAL ACCELEATION A ca is ovi with speed 0 /sec on a cicula path of adius 500. Its speed is inceasi at the ate of /sec. What is the acceleation of the ca at that oent? B. C. A ass is kept on a hoizontal fictionless sufac It is attached to a sti and otates about a fixed cente at an aula velocity 0. If the leth of the sti and aula velocity ae doubled, find the tension in the sti which was initially T 0. A siple pendulu oscillates in a vetical plan When it passes though the ean position, the tension in the sti is ties the weight of the pendulu bob. What is the axiu displaceent of the pendulu of the sti with espect to the vetical. C. SECTION : ADIUS OF CUVATUE AND DYNAMICS OF CICULA MOTION What is the adius of cuvatue of the paabola taced out by the poctile in the pevious poble in the pevious poble at a point whee the paticle velocity akes an ale / with the hoizontal? C 4. A ceili fan has a diaete (of the cicle though the oute edges of the thee blades) of 0 c and p 500 at full speed. Conside a paticle of ass g sticki at the oute end of a blad How uch focedoes it expeience when the fan uns at full speed? Who exets this foce on the paticle? How uch foce does the paticle exet on the blade alo its suface? a C. SECTION : CICULA MOTION IN HOIZONTAL PLANE A sti beaks if its tension exceeds 0 newtons. A stone of ass 50 g tied to this sti of leth 0 c is otated in a hoizontal cicl Find the axiu aula velocity of otation. D. A body of ass has at one end of a sti of leth a, the othe end of which is fixed. It is given a hoizontal velocity u at its lowest position so that the sti would just bee slack, when it akes an ale of 60 with the upwad dawn vetical lin Find the tension in the sti at point of poction D. A ball is ovi to and fo about the lowest point A of a sooth heispheical bowl. If it is able to ise up to a height of 0 c on eithe side of A, find its speed at A. (Take = 0 /s, ass of the body 5 g). D 4. A otocyclist wants to dive on the vetical suface of wooden well of adius 5, with a iniu speed of D. 5 5 /s. Find the iniu value of efficient of fiction between the tyes and the wall of the well. (take g = 0 /s)

2 SECTION (E) : CICULA MOTION IN VETICAL PLANE E. E. E. A ll body of ass is allowed to slide on an inclined fictionless tack fo est position as shown in the figu (i) Find the iniu height h, so that body ay successfully plete the loop of adius. (ii) If h is double of that iniu height, find the esultant foce on the block at position H A nail is located at a cetain distance vetically below the point of suspension of a siple pendulu. The pendulu bob is eleased fo the position whee the sti akes an ale of 60 fo the vetical. Calculate the distance of the nail fo the point of suspension such that the bob will just pefo evolution with the nail as cent Assue the leth of pendulu to be. A bucket tied at the end of a.6 lo sti is whiled in a vetical cicle with nstant speed. What should be the iniu speed so that the wate fo the bucket does not spill, when the bucket is at the highest position (Take g = 0 /sec ) SECTION (F) : MOTION OF A VEHICLE, CENTIFUGAL FOCE AND OTATION OF EATH A tain has to negotiate a cuve of adius 400. By how uch height should the oute ail be aised with espect to inne ail fo a speed of 48 k/h? The distance between the ails is : F. A paticle of ass is nstained to ove alo a goove which is bei otated about a vetical axis though its cente with a nstant aula velocity. If it stats at a distance fo the axis at t = 0, find its velocity elative to the goove when it is at a distance fo the cente F. A pak has a adius of 0. If a vehicle goes ound it at an aveage speed of 8 k/h, what should be the pope ale of banki? F 4. If the oad of the pevious poble is hoizontal (no banki), what should be the iniu fiction a F. efficient so that a stte goi at 8 k/h does not skid. A peson stands on a spi balance at the equato. By what faction is the balance eadi less than his tue weight? If the speed of eath s otation is inceased by such an aount that the balance eadi is half the tue weight, what will be the leth of the day in this case? F 6. A tun of adius 0 is banked fo the vehicles goi at a speed of 6 k/h. If the efficient of static fiction between the oad and the tye is 0.4, what ae the possible speeds of a vehicle so that is neithe slips down no skids up? F 5. SECTION : KINEMATICS OF CICULA MOTION A. The aveage acceleation vecto fo a paticle havi a unifo cicula otion in one plete evolution: A nstant vecto of agnitude v v in agnitude and pependicula to the plane of cicle Equal to the instantaneous acceleation vecto at the stat of the otion A null vecto

3 A. A. A 4. 0 with nstant taential acceleation. It the velocity of the A paticle oves alo a cicle of adius paticle is 80 /s at the end of the send evolution afte otion has begun, the taential acceleation is: 60 /s 40 /s 40 /s 640 /s When a paticle oves in a cicle with a unifo speed its velocity and acceleation ae both nstant its velocity is nstant but the acceleation chaes its acceleation is nstant but the velocity chaes its velocity and acceleation both chae An obct follows a cuved path. The followi quantities ay eain nstant dui the otion speed velocity acceleation agnitude of acceleation Assue that the eath goes ound the sun in a cicula obit with a nstant speed of 0 k/s. The aveage velocity of the eath fo st Jan, 90 to 0th June, 90 is zeo The aveage acceleation dui the above peiod is 60 k/s. The aveage speed fo st Jan, 90 to st Dec, 90 is zeo. The instantaneous acceleation of the eath points towads the sun. A 6. The position vecto of a paticle in a cicula otion about the oigin sweeps out equal aea in equal ti Its velocity eains nstant speed eains nstant acceleation eains nstant taential acceleation eains nstant A 5. SECTION : ADIAL AND TANGENTIAL ACCELEATION Two paticles P and Q ae located at distances P and Q espectively fo the cente of a otati disc such that P > Q : Both P and Q have the e acceleation Both P and Q do not have any acceleation P has geate acceleation than Q Q has geate acceleation than P B. In unifo cicula otion, the quantity that eains nstant is : linea velocity centipetal foce acceleation speed B. When a paticle oves in a cicle with a unifo speed : Its velocity and acceleation ae both nstant Its velocity is nstant but the acceleation chaes Its acceleation is nstant but the velocity chaes Its velocity and acceleation both chae B 4. Fo a paticle in a unifoly acceleated (speed inceasi unifoly) cicula otion : velocity is adial and acceleation is taential only velocity is taential and acceleation is adial only velocity is adial and acceleation has both adial and taential ponents velocity is taential and acceleation has both adial and taential ponents B 5. A oto cyclist goi ound in a cicula tack at nstant speed has nstant linea velocity nstant acceleation nstant aula velocity nstant foce B 6. Fo a body in cicula otion with a nstant aula velocity, the agnitude of the aveage acceleation ove a peiod of half a evolution is...ties the agnitude of its instantaneous acceleation. a B. B 7. A ca of ass oves in a hoizontal cicula path of adius et At an instant its speed is v /s and is inceasi at a ate a /s, then the acceleation of the ca is : v a v a v a

4 B 8. The send s hand of a watch has leth 6 c. Speed of end point and agnitude of diffeence of velocities at two pependicula positions will be : 6.8 & 0 /s 8.88 & 4.44 /s 8.88 & 6.8 /s 6.8 & 8.88 /s B 9. A paticle is goi in a spial path as shown in figue with nstant speed. The velocity of the paticle is nstant The acceleation of the paticle is nstant The agnitude of accleation is nstant The agnitude of accleation is deceasi ntinuously. SECTION : ADIUS OF CUVATUE AND DYNAMICS OF CICULA MOTION C. A paticle of ass is executi unifo cicula otion on a path of adius. If p is the agnitude of its linea oentu. The adial foce acti on the paticle is : A tain is ovi towads noth. At one place it tuns towads noth-east, hee we obseve that the adius of cuvatue of oute ail will be geate than that of the inne ail the adius of the inne ail will be geate than that of the oute ail the adius of cuvatue of one of the ails will be geate The adius of cuvatue of the oute and inne ails will be the e C. p p p p SECTION : CICULA MOTION IN HOIZONTAL PLANE A stone of ass of 6 kg is attached to a sti 44 lo and is whiled in a hoizontal cicl The axiu tension the sti can withstand is 6 newton. The axiu speed of evolution of the stone without beaki it, will be : 0 s 6 s 4 s s D. Thee identical paticles ae joined togethe by a thead as shown in figu All the thee paticles ae ovi on a sooth hoizontal plane about point O. If the velocity of the outeost paticle is v 0, then the atio of tensions in the thee sections of the sti is : : 5 : 7 a D. : 4 : 5 7 : : 6 : 5 : 6 D. A paticle is kept fixed on a tunatable otati unifoly. As seen fo the gound, the paticle goes in a cicle, its speed is 0 c/s and acceleation is 0 c/s.the paticle is now shifted to a new position to ake the adius half of the oiginal valuthe new values of the speed and acceleation will be 0 c/s, 0 c/s 0 c/s, 80 c/s 40 c/s, 0 c/s 40 c/s,40 c/s D 4. A in placed on a otati tuntable just slips if is placed at a distance of 4 c fo the cent if the aula velocity of the tuntable is doubled, it will just slip at a distance of c c 4 c 8 c D 5. A od of leth L is pivoted at one end and is otated with a unifo aula velocity in a hoizontal plane. Let T and T be the tensions at the points L/4 and L/4 away fo the pivoted ends. T > T T > T T = T The elation between T and T depends on whethe the od otates clockwise o anticlockwise SECTION (E) : CICULA MOTION IN VETICAL PLANE E. When a paticle is otated in a vetical plane with nstant aula velocity, agnitude of centipetal foce is :

5 E. Maxiu at highest point Maxiu at lowest point Sae at all points zeo A otocycle is goi on an ovebidge of adius. The dive aintains a nstant speed. As the otocycle is ascendi on the ovebidge, the noal foce on it : incease deceases eains nstant fist inceases then deceases. E. A heavy paticle hai vetically fo a point by a light inextensible sti of leth l is stated so as to ake a plete evolution in a vetical plan The su of the agnitude of tension at the ends of any diaete : fist incease then deceases is nstant fist decease then inceases deceases ntinuously E 4. In a cicus stuntan ides a otobike in a cicula tack of adius in the vetical plan The iniu speed at highest point of tack will be : g g g g A heavy ass is attached to a thin wie and is whiled in a vetical cicl The wie is ost likely to beak when the ass is at the highest point of the cicle when the ass is at the lowest point of the cicle when the wie is hoizontal at an ale of s (/) fo the upwad vetical E 6. A paticle is ovi in a vetical cicl The tensions in the sti when passi though two positions at ales 0 and 60 fo vetical (lowest positions) ae T and T espectively. Then T = T T > T T > T Tension in the sti always eains the e E 7. A ca oves at a nstant speed on a oad as shown in figu The noal foce by the oad on the ca in NA and NB when when it is at the points A and B espectively. g = v g is not geate than E 9. NA < NB insufficient Wate in a bucket is whiled in a vetical cicle with a sti attached to it.the wate does not fall down even when the bucket is inveted at the top of its path. We nclude that in this position. E 8. NA > NB NA = NB a E 5. g is geate than v g is not less than v v A siple pendulu havi a bob of ass is suspended fo the ceili of a ca used in a stunt fil shotti. The ca oves up alo an inclined cliff at a speed v and akes a jup to leave the cliff and lands at soe distance. Let be the axiu height of the ca fo the top of the cliff. The tension is the sti when the ca is in ai is g g v g + v zeo SECTION (F) : MOTION OF A VEHICLE, CENTIFUGAL FOCE AND OTATION OF EATH F. The dive of a ca tavelli at speed suddenly sees a wall at a distance diectly in font of hi. To avoid llision, he should apply bakes shaply he should tun the ca shaply he should apply bakes and then shaply tun None of these

6 F. A cuved section of a oad is banked fo a speed v. If thee is no fiction between oad and tyes of the ca, then: ca is oe likely to slip at speeds highe than v than speeds lowe than v ca cannot eain in static equilibiu on the cuved section ca will not slip when ovi with speed v none of the above F. A paticle of ass is obseved fo an inetial fae of efeence and is found to ove in a cicle of adius with a unifo speed v. The centifugal foce on it is v towads the cente v away fo the cente v alo the taent though the paticle zeo A tain A uns fo east to west and anothe tain B of the e ass uns fo west to east at the e speed alo the equato. A pesses the tack with a foce F and B pesses the tack with a foce F. F > F F < F F= F the infoation is insufficient to find the elation between F and F. F 5. A ca of ass M is ovi hoizontaly on a cicula path of adius. At an instant its speed is v and is inceasi at a ate a. The acceleation of the ca is towads the cente of the path F 4. v (C*) The fiction efficient between the gound and the ca is not less than a/g. (B*) The agnitude of the fictional foce on the ca is geate than v g a A cicula oad of adius is banked fo a speed v = 40 k/h. A ca of ass attepts to go on the cicula oad. The fiction efficient between the tye and the oad is negligibl The ca cannot ake a tun without skiddi. (B*) If the ca tuns at a speed less than 40 k/h, it will slip down F 6. The fiction efficient between the gound and the ca is = tan v (D*) If the ca tuns at the ect speed of 40 k/h, the foce by the oad on the ca is geate than g as If the ca tuns at the cuent speed of 40 k/h, the foce by the oad on the ca is equal to well as geate than F 7. v A peson applies a nstant foce F on a paticle of ass and finds that the paticle oves in a cicle of adius with a unifo speed v as seen fo an inetial fae of efeenc This is not possibl Thee ae othe foces on the paticle v towads the cent The esultant of the othe foces vaies in agnitude as well as in diection. The esultant of the othe foces is. A paticle of ass begins to slide down a fixed sooth sphee fo the top. What is its taential acceleation when it beaks off the sphee? g 5g g g

7 . A kg stone at the end of lo sti is whiled in a vetical cicle at nstant speed of 4 /sec. The tension in the sti is 6 N when the stone is at (g = 0 /sec) : top of the cicle botto of the cicle halfway down none of these. A boy whils a stone in a hoizontal cicle.8 above the gound by eans of a sti. lo. The sti beaks and stone flies off hoizontally, stiki the gound 9. away. The centipetal acceleation dui the cicula otion was: (use g = 9.8 /s) 94 /s 4 /s 88 /s 8 /s 4. A paticle of ass is suspended fo a fixed point O by a sti of leth. At t = 0, it is displaced fo equilibiu position and eleased. The gaph, which shows the vaiation of the tension T in the sti with tie t, ay be : A paticle P is ovi in a cicle of adius a with a unifo speed u. C is the cente of the cicle and AB is diaet The aula velocity of P about A and C ae in the atio : : : : 4 : 6. The distance of the two planets fo the sun ae nealy 0 to 0 etes espectively. Then the atio of thei peiodic tie, if they ae supposed to ove in a cicula obit is : : : 0 0 0, whili in a cicula path in a vetical plan The atio of axiu tension in the sti to the iniu tension in the sti is 4, If g is taken to be 0 /s, the speed of the stone at the highest point of the cicle is : a A stone of ass kg tied to a light inextensible sti of leth L = 0 /s 5 /s 0 /s 0 /s Toy cat tied of the end of an unstetched sti of leth a, when evolved oves in a hoizontal cicle of adius a with a tie peiod T. Now the toy cat is speeded up until it oves in a hoizontal cicle of adius a with a peiod T. If Hook s law holds then : 8. : 7. 0 : 5. T = T T = T T = T A ll sphee of ass suspended by a thead is fist taken aside so that the thead fos the ight ale with the vetical and then eleased, then : (i) The total acceleation of the sphee and the thead tension as a function of, the ale of deflection of the thead fo the vetical will be T = 9. T (A*) g s, T = g s g s, T = g s. g sin, T = 5g s g sin, T = 5 g s. (ii) The thead tension at the oent when the vetical ponent of the sphee s velocity is axiu will be g g g (C*) g (iii) The ale between the thead and the vetical at the oent when the total acceleation

8 vecto of the sphee is diected hoizontally will be s = 0. sin = sin = The kinetic enegy k of a paticle ovi alo a cicle of adius depends on the distance veed s as k = as whee a is a nstant. The total foce acti on the paticle is : / s as s a. s = as a s Two ll sphees, each of ass ae igidly nnected by a od of negligible ass and ae eleased fo est in the position shown and slide down a sooth cicula guide in the vetical plan y x 45 A g 0 g g g g g / (i) Thei on velocity as they each the hoizontal dashed position will be : A paticle oves with deceleation alo the cicle of adius so that at any oent of tie its taential and noal acceleations ae equal in oduli. At the initial oent t = 0 the speed of the paticle equals v0, then : (i) the speed of the paticle as a function of the distance veed s will be v = v0 e s/ v = v0 es/ v = v0 e /s v = v0 e/s. a (ii) The foce between sphee and the suppoti suface at an instant just befoe the sphee eaches the botto position A will be :.9 g.9 g g g/ a =. v (ii) the total acceleation of the paticle as function of velocity and distance veed a = v a = v a = v A paticle A oves alo a cicle of adius = 50 c so that its adius vecto elative to the point O (Fig.) otates with the nstant aula velocity = 0.40 ad/s. Then odulus of the velocity of the paticle, and the odulus of its total acceleation will be v = 0.4 /s, w = 0.4 /s v = 0. /s, w = 0.4 /s 4. v = 0. /s, w = 0. /s v = 0.4 /s, w = 0. /s A spot light S otates in a hoizontal plane with a nstant aula velocity of 0. ad/s. The spot of light P oves alo the wall at a distance. What is the velocity of the spot P when = 450? 0.6 /s 0.5 /s 0.4 /s 0. /s

9 . A sooth od PQ otates in a hoizontal plane about its id point M which is h = 0. vetically below a fixed point A at a nstant aula velocity 4 ad/s. A light elastic sti of natual leth 0. equii.47 N/c has one end fixed at A and its othe end attached to a i of ass = 0. kg which is fee to slide alo the od. When the i is stationay elative to od, then inclination of sti with vetical, tension in sti, foce exeted by i on the od will be s = /5, T = 9.8 N, N =.88 N s = /5, T = 4.9 N, N =.44 N = 60, T = 0, N =.44 N = 0, T = 0, N =.88 N Figue shows the total acceleation and velocity of a paticle ovi clockwise in a cicle of adius.5 at a given instant of ti At this instant, find : (a) the adial acceleation, (b) the speed of the paticle and (c) its taential acceleation. Two paticle A and B ove anticlockwise with the e speed v in a cicle of adius and ae diaetically. 7v. Calculate the tie 5 in which A llides with B, the ale taced by A, its aula velocity and adial acceleation at the tie of llision. A otocycle has to ove with a nstant speed on an ovebidge which is in the fo of a cicula ac of adius and has a total leth L. Suppose the otocycle stats fo the highest point. (a) What can its axiu velocity be fo which the ntact with the oad is not boken at the highest point. a 4. opposite to each oth At t = 0, A is given a nstant acceleation (taential) at = A 4 kg block is attached to a vetical od by eans of two stis of equal leth. When the syste otates about the axis of the od, the stis ae extended as shown in figu 5. (b) If the otocycle goes at speed / ties the axiu found in pat (a) whee will it lose the ntact with the oad. (c) What axiu unifo speed can it aintain on the bidge if it does not lose ntact anywhee on the bidge? (a) How any evolutions pe inute ust the syste ake in ode fo the tension in the uppe chod to be 0 kgf? (b) What is the tension in the lowe chod then? 6. A etallic chain with a leth and whose ends ae joined togethe is fitted onto a wooden disc as shown in the figu The disc otates with a speed of n evolutions pe send. Find the tension of the chain T if its ass is.

10 7. A ll obct of ass is tied to a sti of leth l and is whiled ound in a hoizontal cicle of adius at nstant speed v (figue). The cente of the cicle is vetically below the point of suppot. [This aaeent is called nical pendulu.] Calculate v in tes of g, and. Also calculate the peiod of evolution. 8. A ca goes on a hoizontal cicula oad of adius, the speed inceasi at a nstant ate dv = a. The fiction efficient between the oad and the tye is. Find the speed at which the ca will dt skid. A tack nsists of two cicula pats ABC and CDE of equal adius 00 and joined soothly as shown in fig. Each pat subtends a ight ale at its cent A cycle weighi 00 kg togethe with the ide tavels at a nstant speed of 8 k/h on the ack. Find the noal ntact foce by the oad on the cycle when it is at B and D. Find the ce of fiction exeted by the tack on the tyes when the cycle is at B, C and D. (c) Find the noal foce between the oad and the cycle just, befoe and just afte the cycle sses C. What should be the iniu fiction efficient between the oad and the tye, which will ensue that the cyclist can ove with nstant speed? Take g = 0/s. 0. A block of ass oves on a hoizontal cicle against the wall of a cylindical oo of adius. The floo of the oo on which the block oves is sooth but the fiction efficient between the wall and the block is µ. The block is given an initial speed v 0. As a function of the instantaneous speed v wite the noal foce by the wall on the block, the fictional foce by the wall and the taential acceleation of the block. obtain the speed of the block afte one evolution.. A table with sooth hoizontal suface is fixed in a cabin that otates with a unifo aula velocity in a cicula path of adius (figue). A sooth goove AB of leth L(< < ) is ade on the suface of the table.the goove akes an ale with the adius OA of the cicle in which the cabin otates. A ll paticle is kept at the point A in the goove and is eleased to ove alo AB. Find the tie taken by the paticle to each the point B.. A ca ovi at a speed of 6 k/h is taki a tun on a cicula oad of adius 50. A ll wodden plate is kept on the seat with its plane pependicula to the adius of the cicula oad figu A ll block of ass 00g is kept on the seat which ests against the plat The fiction efficient between the block and the plate is = Find the noal ntact foce exeted by the plate on the block. The plate is slowly tuned so that the ale between the noal to the plate and adius of the ale a 9.

11 between the noal to the plate and the adius of the oad slowly inceases. Find the ale at which the block will just stat slidi on the plat A table with sooth hoizontal suface is placed in a cabin which oves in a cicle of a lage adius (figue). A sooth pulley of ll adius is fastended to the tabl Two asses and placed on the table ae nneted though a sti ove the pulley. Initially the asses ae held by a peson with the sti alo the outwad adius and then the syste is eleased fo est (with espect to the cabin). Find the agnitude of the initial acceleation of the asses as seen fo the cabin and the tension in the sti.. A heispheical bowl of adius = 0. is otati about its axis (which is vetical) with an aula velocity. A paticle of ass 0-kg on the fictionless inne suface of the bowl is also otati with the e. the paticle is at a height h fo the botto of the bowl. (a) Obtain the elation between h and. What is the iniu value of needed in ode to have a nonzeo value of h. (b) It is desied to easue g usi this setup by easui h accuately. Assui that and ae known pecisely and that the least unt in the easueent of h is 0-4. What is iniu eo g in the easued value of g.[g = 9.8/s] [JEE 99] A paticle of ass is ovi in a cicula path of nstant adius such that is centipetal acceleation ac is vayi with tie t as ac = k t whee k is a nstant. The powe deliveed to the paticle by the foce acti on it is[jee 994] a.. (k 4 t 5 ) Zeo A sooth seicicula wie tack of adius is fixed in a vetical plane (figue). On end of a assless spi of natual leth /4 is attached to the lowest point O of the wie tack. A ll i of ass which can slide on the tack is attached to the othe end of the spi. The i is held stationay at point P such that the spi ake an ale 60 with the vetical. The spi nstant K = g/. Conside the instant when the i is eleased (i) Daw the fee body diaga of the i. (ii) Deteine the taential acceleation of the i and the noal eaction. [JEE 996] k. 4. k t Two blocks of ass = 0 Kg and = 5 Kg nnected to each othe by a assless inextensible sti of leth 0. ae placed alo a diaete of tun tabl The efficient of fiction between the table and is 0.5 while thee is no fiction between and the tabl The table is otati with an aula velocity of 0 ad/s about a vetical axis passi though its cente O. The asses ae placed alo the diaete of the table on eithe side of the cente O such that the ass is at a distance of 0.4 fo O. The asses ae obseved to be at est with espect to an obseve on the tun tabl (i) Calculate the fictional foce on. (ii) What should be the iniu aula speed of the tun table so that the asses will slip

12 fo this position. How should the asses by placed with the sti eaini taut so that thee is no fictional foce acti on the ass. [JEE 997] A ll block of ass slides alo a sooth fictional tack as shown in figu (i) If it stats fo est at P, when is the esultant foce acti on it at Q? (ii) At what height above the botto of the loop should the block be eleased so that the foce it exets against the tack at the top of the loop equals its weight? (iii) [EE 997] A stone tied to a sti of leth L is whiled in a vetical cicle with the othe end of the sti at the cent At a cetain instant of tie the stone is at its lowest position and has a speed u. The agnitude of the chae in its velocity as it eaches a position, whee the sti is hoizontal, is [JEE 998] u gl gl u gl (u gl ) A paticle at est stats olli fo the top of a lage fictionless sphee of adius. The sphee is fixed on the gound. Calculate that height fo the gound at which the paticle leaves the suface of the sphe [EE 998] 8. A paticle is suspended vetically fo a point O by an inextensible assless sti of leth L. A vetical line AB is at a distance L/8 fo O as shown. The obct given a hoizontal velocity u. At soe point, its otion ceases to be cicula and eventually the obct passes though the line AB. At the instant of ssi AB, its velocity is hoizontal. Find u. 7. a O A L/8 L A lo hoizontal od has a bead which can slide alo its leth and is initially placed at a distance L fo one end A of the od. The od is set in aula otion about A with a nstant aula acceleation,. If the efficient of fiction between the od and the bead is, and gavity is neglected, then the tie afte which the bead stats slippi is [JEE 000] 0. Infinitesial A ll block is shot into each of the fou tacks as shown below. Each of the tacks ises to the e height. The speed with which the block entes the tack is the e in all cases. At the highest point of the tack, the noal eaction is axiu is[jee 00]. 9. B u [JEE 999] An insect cawls up a heispheical suface vey slowly (see the figue). The efficient of fiction between the suface and the insect is /. If the line joini the cente of the heispheical suface to the insect akes an ale with the vetical, the axiu possible value of is given t = tan = sec = sec = [JEE 00]

13 . A block is placed inside a hoizontal hollow cylind The cylinde stats otati with one evolution pe send about its axis. The aula position of the block at which it begins to slide is 0 below the hoizontal level passi though the cent Find the adius of the cylinde if the efficient of fiction is 0.6. What should be the iniu aula speed of the cylinde so that the block each the highest point of the cylinde? [EE 00]. A siple pendulu is oscillati without dapi. When the displaceents of the bob is less than axi u, its acceleation vecto a is ectly shown in [JEE 00] A spheical ball of ass is kept at the highest point in the space between two fixed, ncentic sphees A and B (see figue). The lle sphee A has a adius and the space between the two sphees has a width d. The ball has a diaete vey slightly less than d. All sufaces ae fictionless. The ball given a gentle push (towads the ight in the figue). The ale ade by the adius vecto of the ball with the upwad vetical is depended by (shown in figue) (a) Expess the total noal eaction foce exeted by the sphees on the ball as a function of ale. (b) Let NA and NB denote the agnitudes of the noal eaction foce on the ball extoted by the sphees A and B, espectively. Sketch the vaiations of NA and NB as functions of s in the ae 0 by dawi two sepaate gaphs in you answe book, taki s on the hoizontal axis q Sphee O ¼xksy k½b d 4. ¼xksy k½a TA TB A B A double sta syste nsists of two stas A and B which have tie peiod TA and TB. adius A and B and ass MA and MB. Choose the ect option. [IIT 006, (, ] If TA > TB then A > B If TA > TB then MA > MB 5. a Sphee [JEE 00] TA = TB

14 EXECISE # SECTION : KINEMATICS OF CICULA MOTION A. (b) fist (b) thid. S E C TI O N (B ) : AD I AL ACCELEATION B..7 /sec AND TA NG E NT IA L S E C TI ON (C ) : AD IU S O F C U VATU E AN D DYNAMICS OF CICULA MOTION C. 8 T 0 C. 90 C 4. g s ( / ) F. C F 5. BC F. D F 6. BD 5 (ii) F = 6 g E A 5. B 8. B (ii) C. C 6. D (ii) B (ii) A 4. A EXECISE # 4. A. (a).65 /s (b) 7.5 /s (c).5 /s 5 7v 89 v,,, 5 6v c F. v = tan (/4)] F ,.0 hou] Between 4.7 k/h and 54 k/h] EXECISE #. 4. SECTION : KINEMATICS OF CICULA MOTION A. D A. C A. D A 4. AD A 5. D A 6. BD S E C TI O N (B ) : AD I AL ACCELEATION B. C B. D B 4. D B 5. C B 7. B B 8. D AND TA NG E NT IA L B. D B 6. A B 9. C (c) v = g, L g s 5. (a) 9.6 pe in., (b) 50 N 6. T = n 7. g tan, g tan 8. Net foce on ca = fictional foce f a SECTION : CICULA MOTION IN HOIZONTAL PLANE D. D D. D D. A D 4. A D 5. A f= v4 (whee is ass of the ca)...() Fo skiddi to just occu f = µn = µg...() Fo () and () S E C TI ON (C ) : AD IU S O F C U VATU E AN D DYNAMICS OF CICULA MOTION C. D C. A SE CTION (E ) : CICULA MOTION IN V E TICAL (a) vax = (b) At ale = 60º fo the vetical position. F. F. F 5. F 6. a S E C TI ON (F ) : M OT IO N O F A V E H IC LE, CE NTIFUG AL FOCE AND OTATION OF EATH (iii) A. B 4. D 7. A 9. (i) A 0. B. (i) A. (i) A. D 4 /sec F. A F 4. A F 7. BD E. S E C TI ON (F ) : M OT IO N O F A V E H IC LE, CE NTIFUG AL FOCE AND OTATION OF EATH EXECISE # SE CTION (E ) : CICULA MOTION IN V E TICAL PLANE (i) hin = E. B E 6. C E 9. D [4.8N, 4.8 N] SECTION : CICULA MOTION IN HOIZONTAL PLANE D. 0 ad/s D. 4.5 g D. /s D E. E. A E 5. B E 8. C C. u s PLANE E. C E 4. D E 7. C v = { [ g a ]}/ 4 ( g 9. a ) 4 975N, 05 N, 68N, 7 N, 0,707N, 0, (D ) 0.07]

15 EXECISE # 5. (a) 7 ad / s (b) -9.8 x 0- /s B. (i) 4. (i) 6 N (ii).67 ad/s (iii) = 0. and = The noal eaction by wall on the (ii) v The fiction foce on the block by the (iii) µv wall is f = µn = The taential acceleation of the block is N = v 0 dv µv = ds.. v0. v 4. a (ii) µv o (i) (b) s =/ 5. D L s 0.N, dv µ ds v n v = µ 0 v = v0 e µ (iii) g v integati we get Ans. 6. D 0º (iv) v0e µv 8. u gl. A. C NA = H = (5/) 7. g 5 g, N = (a) N = g s - g, v o 65 g, 9. A 0. A. 0.4,6.4ad/sec f µv = dv µv = dt (iv) block = 5. (i) (ii) at = s = NB 5g s s s =/ s =