NEETIIT.COM. Angular Displacement. Page - 1

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1 - Download ou andoid App. 1. ANGULA DISPLACEMENT Intoduction : Angle subtended by position ecto of a paticle moing along any abitay path w..t. some fixed point is called angula displacement. (a) Paticle moing in an abitay path (b) Paticle moing in staight line (c) Paticle moing in cicula path (i) O Fixed point O O Angula displacement is a ecto quantity. (ii) Its diection is pependicula to plane of otation and gien by ight hand scew ule. Note: Clockwise angula displacement is taken as negatie and anticlockwise displacement as positie. ac linea displacement angle adius adius (iii) Fo cicula motion S (i) Its unit is adian (in M.K.S) Note : Always change degee into adian, if it occus in numeical poblems. 60 o Note : 1 adian adian 180º () It is a dimensionless quantity i.e. dimension [M 0 L 0 T 0 ] Q Q P Q P S P Ex.1 Angula Displacement A paticle completes 1.5 eolutions in a cicula path of adius cm. The angula displacement of the paticle will be - (in adian) (A) 6 (B) (C) (D) Sol.(D) We hae angula displacement lineadisplacement adiusof path S Hee, S n() 1.5 ( 10 ) Hence coect answe is (B). ANGULA VELOCITY adian It is defined as the ate of change of angula displacement of a body o paticle moing in cicula path. (i) It is a ecto quantity. (ii) Its diection is same as that of angula displacement i.e. pependicula to plane of otation. Note : If the paticle is eoling in the clockwise diection then the diection of angula elocity is pependicula to the plane downwads. Wheeas in case of anticlockwise diection the diection will be upwads. (iii) Its unit is adian/sec (i) Its dimension is [M 0 L 0 T 1 ] Types of Angula Velocity :.1 Aeage Angula Velocity : Total angula displacement a Total time taken. Instantaneous Angula elocity : Note: The intantaneous angula elocity is defined as the angula elocity at some paticula instant. Instantaneous angula elocity lim t0 t d Instantaneous angula elocity can also be called as simply angula elocity. - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page - 1

2 - Download ou andoid App. Ex. Aeage Angula Velocity A paticle eoling in a cicula path completes fist one thid of cicumfeence in sec, while next one thid in 1 sec. The aeage angula elocity of paticle will be : (in ad/sec) (A) (C) 4 (B) (D) 5 Total angula displacement Sol.(A) We hae a Total time Ex. Fo fist one thid pat of cicle, angula displacement, S 1 / 1 Fo second one thid pat of cicle, / Total angula displacement, Total time ad 1 + 4/ ad + 1 sec 4 / a ad/s 4 ad/s 6 Hence coect answe is (A) The atio of angula speeds of minute hand and hou hand of a watch is - (A) 1 : 1 (B) 6 : 1 (C) 1 : 1 (D) 1 : 6 Sol.(C) Angula speed of hou hand, 1 t 1 60 ad/sec angula speed of minute hand, 1 ad/sec Hence coect answe is (C). Ex.4 Instantaneous Angula Velocity The angula displacement of a paticle is gien by 0 t + 1 t, whee 0 and ae constant and 0 1 ad/sec, 1.5 ad/sec. The angula elocity at time, t sec will be (in ad/sec) - (A) 1 (B) 5 (C) (D) 4 Sol.(D) We hae 0 t + 1 t d 0 + t This is angula elocity at time t. Now angula elocity at t sec will be d 0 + tsec 1 + x ad/sec Hence coect answe is (D). ELATION BETWEEN LINEA VELOCITY AND ANGULA VELOCITY Note : d d ds We hae ds 1. ds ac [ d, angle d adius ds and linea elocity] In ecto fom, (i) When a paticle moes along a cued path, its linea elocity at a point is along the tangent dawn at that point (ii) When a paticle moes along cued path, its elocity has two components. One along the adius, which inceases o deceases the adius and anothe one pependicula to the adius, which makes the paticle to eole about the point of obseation. - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page -

3 - Download ou andoid App. Ex.5 Sol. (iii) t sin Linea Velocity & Angula Velocity A paticle moes in a cicle of adius 0cm with a linea speed of 10m/s. The angula elocity will be - (A) 50 ad/s (B) 100 ad/s (C) 5 ad/s (D) 75 ad/s The angula elocity is Hence 10 m/s 0 cm 0. m, 50 ad/s Hence coect answe is (A) 4. ANGULA ACCELEATION The ate of change of angula elocity is defined as angula acceleation. If be change in angula elocity in time t, then angula acceleation lim t0 t d (i) It is a ecto quantity (ii) Its diection is that of change in angula elocity (iii) Unit : ad/sec (i) Dimension : M 0 L 0 T Ex.6 elation Between Angula Velocity & Angula Acceleation The angula elocity of a paticle is gien by 1.5 t t +, the time when its angula acceleation deceases to be zeo will be - (A) 5 sec (B) 0.5 sec (C) 1 sec (D) 1. sec Sol.(B) Gien that 1.5t t + d 1.5 6t When t t sec Hence coect answe is (B) 5. ELATION BETWEEN ANGULA ACCELEATION AND LINEAACCELEATION Linea acceleation ate of change of linea elocity d a...(i) Angula acceleation ate of change of angula elocity Fom (i) & (ii) In ecto fom Ex.7 Sol.(C) Gien d...(ii) a d d() d d d [ is constant] d a a elation Between Angula Acceleation & Linea Acceleation A paticle is moing in a cicula path with elocity aying with time as 1.5t + t. If cm the adius of cicula path, the angula acceleation at t sec will be - (A) 4 ad/sec (B) 40 ad/sec (C) 400 ad/sec (D) 0.4 ad/sec Linea acceleation a 1.5 t + t d t + This is the linea acceleation at time t Now angula acceleation at time t a t 10 Angula acceleation at t sec 8 () at t sec ad/sec Hence coect answe is (C) - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page -

4 - Download ou andoid App. 6. EQUAT ION OF LINEA MOT ION AND OTATIONAL MOTION (i) With constant elocity a 0, s ut 0, t (ii) With constant acceleation (i) Aeage elocity (i) Aeage angula elocity Ex.8 a Equations of otational Motion A gind stone stats f om est and has a constant-angula acceleation of 4.0 ad/sec.the angula displacement and angula elocity, afte 4 sec. will espectiely be - (A) ad, 16 ad/sec (B) 16ad, ad/s (C) 64ad, ad/sec (D) ad, 64ad/sec u (ii) Aeage acceleation a a u t (iii) s a t (i) u + at () s ut + (i) s t (iii) With aiable acceleation (i) a 1 (ii) Aeage angula acceleation a a 1 t u t (iii) a. t (i) 1 + t 1 1 at () 1 t + t 1 at (i) t 1 t (ii) u + as (i) 1 + (iii) S n u + 1 (n 1)a displacement in n th sec ds (ii) ds (iii) a d (i) d a () d a ds Sol. d ds 1 (iii) n (n 1) Angula displacement in n th sec (i) d/ (ii) d (iii) d d d (i) d () d d Angula displacement afte 4 sec is 0 t + 1 t 1 t ad Angula elocity afte 4 sec Hence coect answe is (A) 0 + t ad/sec t - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page - 4

5 - Download ou andoid App. elation Between Angula Velocity 7.1 Expession fo Centipetal Acceleation & Angula Acceleation Ex.9 The shaft of an electic moto stats fom est and on the application of a toque, it gains an angula acceleation gien by t t duing the fist seconds afte it stats afte which 0. The angula elocity afte 6 sec will be - (A) 10/ ad/sec (C) 0/4 ad/sec Sol.(A) Gien t t d t t d (t t ) (B) /10 ad/sec (D) 4/0 ad/sec t t c at t 0, 0 c 0, Angula elocity at t t t sec, t sec 8 10 (4) ad/sec Since thee is no angula acceleation afte sec The angula elocity afte 6 sec emains the same. Hence coect answe is (A) 7. CENTIPETAL ACCELEATION AND CENTIPETAL FOCE (i) A body o paticle moing in a cued path always moes effectiely in a cicle at any instant. (ii) The elocity of the paticle changes moing on the cued path, this change in elocity is bought by a foce known as centipetal foce and the acceleation so poduced in the body is known as centipetal acceleation. (iii) The diection of centipetal f oce o acceleation is always towads the cente of cicula path. (a) Paticle moing (b) Vecto diagam of in cicula path of elocities adius (i) The tiangle OP 1 P and the elocity tiangle ae simila P1 P AB P1 O AQ s [ 1 ] s s t t lim t0 t lim s t0 t a c This is the magnitude of centipetal acceleation of paticle It is a ecto quantity. In ecto fom (ii) The diection of that of a c ac would be the same as (iii) Because elocity ecto at any point is tangential to the cicula path at that point, the acceleation ecto acts along adius of the cicle at that point and is diected towads the cente. This is the eason that it is called centipetal acceleation. Ex.10 O P (t + t) P 1 (t) 1 1 Centipetal Acceleation A ball is fixed to the end of a sting and is otated in a hoizontal cicle of adius 5 m with a speed of 10 m/sec. The acceleation of the ball will be - (A) 0 m/s (B) 10 m/s (C) 0 m/s (D) 40 m/s - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page - 5

6 - Download ou andoid App. Sol.(A) We know a Hence 10 m/s, 5 m (10) a 0 m/s 5 Hence coect answe is (A) Calculation of Centipetal Acceleation by Angula Velocity - Linea Velocity elation Ex.11 A body of mass kg lying on a smooth suface is attached to a sting m long and then whiled ound in a hoizontal cicle making 60 eolution pe minute. The centipetal acceleation will be - (A) m/s (B)1.18 m/s (C).68 m/s (D).68 m/s Sol.(A) Gien that the mass of the paticle, m kg adius of cicle m Angula elocity 60 e/minute 60 ad/sec 60 ad/sec Because the angle descibed duing 1 eolution is adian The linea elocity m/s 6 m/s The centipetal acceleation m/s Hence coect answe is (A) 7. Expession fo Centipetal foce If elocity of paticle, adius of path Then necessay centipetal foce mass acceleation m (6) m/s This is the expession fo centipetal foce (i) Note : It is a ecto quantity (ii) In ecto fom. ˆ m ˆ m m ( ) negatie sign indicates diection only (iii) Fo cicula motion : Fc m ( ) Fc O A m ( sin 90º) 1. Centipetal foce is not a eal foce. It is only the equiement fo cicula motion.. It is not a new kind of foce. Any of the foces found in natue such as gaitational foce, electic fiction foce, tension in sting Ex.1 eaction foce may act as centipetal foce. Angula Velocity - Centipetal Foce elation A body of mass 0.1 kg is moing on cicula path of diamete 1.0 m at the ate of 10 eolutions pe 1.4 seconds. The centipetal foce acting on the body is - (A) 0. N (B) 0.4 N (C) N (D) 4 N Sol.(A) F m Hee m 0.10 kg, 0.5 m n.1410 and t 1. 4 ad/s F () 0. Hence coect answe is (A) Centipetal Foce - Angula Velocity elation Ex.1 A body of mass 4 kg is mo ing in a hoizontal cicle of adius 1 m with an angula elocity of ad/s. The equied centipetal foce, will be - (A) 16 N (B) 1.6 N (C) 16 Dyne (D) 1.6 Dyne Sol.(A) F m N Hence coect answe is (A) - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page - 6

7 - Download ou andoid App. Centipetal & Fiction foce elation Ex.14 The safe elocity equied fo scooteist negotiating a cue of adius 00 m on a oad with the angle of epose of tan 1 (0.) will be- (A) 0 km/h (B) 00 m/s (C) 7 km/h (D) 7 m/s Sol.(C) As the centipetal foce is supplied by the fictional foce, hence tan 1 (0.) tan 1 () (0.)] 0 m/s 18 The safe speed is 0 7 km/h 5 Hence coect answe is (C) Centipetal Foce Ex.15 A body of mass 4 kg is tied to one end of a ope of length 40 cm and whiled in a hoizontal cicle. The maximum numbe of eolutions pe minute it can be whiled so that the ope does not snap as the ope can with stand to a tension of 6.4 Newton, will be - (A) 1.91 (B) 19.1 (C) 191 (D) 1910 Sol.(B) Tension in the ope m m 4 n Maximum tension 6.4 N n Numbe of eolutions pe minutes 60/ 19.1 Hence coect answe is (B) Ex.16 A cetain sting which is 1 m long will beak, if the load on it is moe than 0.5 kg. A mass of 0.05 kg is attached to one end of it and the paticle is whiled ound a hoizontal cicle by holding the fee end of the sting by one hand. The geatest numbe of eolutions pe minute possible without beaking the sting will be- (A) 9.45 (B) 94.5 (C) 99.5 (D) 9.95 Sol.(B) Mass of the body m 0.05 kg, adius of cicula path 1 m The maximum tension in the sting can withstand 0.5 kg wt N 4.9 N Hence the centipetal foce equied to poduce the maximum tension in the sting is 4.9 N i.e. m m 98 n 98 n e/sec 94.5 e/min Hence coect answe is (B) TYPE OF CICULA MOTION 8.1 Unifom cicula motion 8. Non Unifom Cicula Motion : 8.1 Unifom Cicula Motion : Note: If m mass of body, adius of cicula obit, magnitude of elocity a c centipetal acceleation, a t tangential acceleation In unifom cicula motion : (i) 1 constant i.e. speed is constant (ii) As is constant so tangential acceleation a t 0 (iii) Tangential foce F t 0 (i) Total acceleation (i) a a a a c c t (towads the cente) Because is always pependicula to elocity o displacement, hence the wok done by this foce will always be zeo. (ii) Cicula motion in hoizontal plane is usually unifom cicula motion. (iii) Thee is an impotant diffeence between the pojectile motion and cicula motion. In pojectile motion, both the magnitude and the diection of acceleation (g) emain constant, while in cicula motion the magnitude emains constant but the diection continuously changes. Hence equations of motion ae not applicable fo cicula motion. emembe that equations of motion emain alid only when both the magnitude & diection of acceleation ae constant. 1 a c at 0 F t 0 - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page - 7

8 - Download ou andoid App Hint to sole numeical poblem : Sol. (i) Wite down the equied centipetal foce (ii) Daw the fee body diagam of each component of system. (iii) esole the foces acting on the otating paticle along adius and pependicula to adius (i) Calculate net adial foce acting towads cente of cicula path. () Make it equal to equied centipetal foce. (i) Fo emaining components see accoding to question. Ex.17 Centipetal Foce A body of mass m is attached with a sting of length l. If it is whiled in a hoizontal cicula path with elocity. The tension in the sting will be - (A) l (B) m (C) (D) Sol.(B) equied centipetal foce, Ex.18 Hee centipetal foce is poided by the tension in the sting T m T Hence coect answe is (B) Obital Velocity of Satellite A satellite of mass m is eoling aound the eath of mass M in cicula obit of adius. The obital elocity of the satellite will be - Note : (i) The equied centipetal foce, F C (towads the cente) Net foce towads the cente, GMm F G (This foce will poide equied centipetal foce) Theefoe F C F G GMm GM Hence coect answe is (A) Fom aboe example we see that obital elocity of a body is independent to its mass (ii) If we ae asked to find out time peiod of abo e body then time peiod can be calculated as Ex.19 T GM T this is Keple's law. Centipetal Foce Thee identical paticles ae connected by thee stings as shown in fig. These paticles ae eoling in a hoizontal plane. The elocity of oute most paticle is. Then T 1 : T : T will be - (Whee T 1 is tension in the oute most sting etc.) O F G O m m m l l l (A) (C) GM GM m (B) (D) Gm Gm M (A) : 5 : 7 (B) : 5 : 6 (C) : 4 : 5 (D) 7 : 5 : - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page - 8

9 - Download ou andoid App. Sol.(B) Fo A : c B A Note: O equied centipetal foce A l (net foce towads cente T 1 ) This will poide equied centipetal foce paticle at A, T 1 Fo B : A l equied centipetal foce m( ) emembe i.e. angula elocity, of all the paticles is same B A B C When a system of paticles otates about an axis, the angula elocity of all the paticles will be same, but thei linea elocity will be diffeent, because of diffeent distances fom axis of otation i.e.. Thus fo B, centipetal foce A 9 Net foce towads the cente T T 1 T (Putting alue of T 1 ) Fo C : Centipetal foce. C l A 9 A + T 9 1 A 9l Net foce towads cente T T T T T C T B T A 1 A 9l 5 A 9l Note: T A + T 9l 6 A T 9l (on putting alue of T ) Now T 1 : T : T 1 : 9 5 : 9 6 : 5 : 6 It is to be pondeed fom the aboe example that as the elocity is inceased continuously, the innemost sting will beak fist i.e. T > T > T 1 Hence coect answe is (B) 8.1. Motion In Hoizontal Cicle : Conical pendulum This is the best example of unifom cicula motion A conical pendulum consists of a body attached to a sting, such that it can eole in a hoizontal cicle with unifom speed. The sting taces out a cone in the space. (i) The foce acting on the bob ae (a) Tension T (b) weight (ii) The hoizontal component T sin of the tension T poides the centipetal foce and the etical component T cos balances the weight of bob T sin and T cos Fom these equation and T 1...(i) g tan g Also if h height of conical pendulum 4...(ii) - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page - 9

10 - Download ou andoid App. Fom (ii) & (iii), OP tan OS h g h The time peiod of eolution T h g [whee OS l] cos g Motion of Paticle in Hoizontal Cicle...(iii) Ex.0 A paticle descibes a hoizontal cicle on the smooth suface of an ineted cone. The height of the plane of the cicle aboe the etex is 9.8 cm. The speed of the paticle will be - (A) 9.8 m/s (B) 0.98 m/s (C) m/s (D) 98 m/s Sol.(B) The foce acting on paticle ae (i) weight acting etically downwad (ii) Nomal eaction N of the smooth suface of the cone. (iii) eaction of the centipetal foce acting adially outwads. esoling N into hoizontal and etical components we obtain N cos and N sin Ex.1 But tan h h g hg m/s Hence coect answe is (B) A sting of length 1 m is fixed at one end and caies a mass of 100 gm at the othe end. The sting makes / eolutions pe second about a etical axis though the fixed end. The angle of inclination of the sting with the etical, and the linea elocity of the mass will espectiely be - (in M.K.S. system) (A) 5º14',.16 (B) 50º14', 1.6 (C) 5º14', 1.6 (D) 50º14',.16 Sol.(A) Let T be the tension, the angle made by the sting with the etical though the point of suspension. The time peiod h 1 t g fequency Theefoe g h 16 1 g 4 h / cos h 16 g º 14' Linea elocity (l sin ) 1 sin 5º 14' 4.16 m/s Hence coect answe is (A) 8. Non-unifom Cicula Motion : (i) In non-unifom cicula motion : constant constant i.e. speed constant i.e. angula elocity constant h T Nsin Ncos tan g / (ii) If at any instant magnitude of elocity of paticle adius of cicula path angula elocity of paticle, then - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page - 10

11 - Download ou andoid App. (iii) Tangential acceleation : d a t whee (i) Tangential foce : () Centipetal foce : ds and s ac - length F t ma t (i) Net foce on the paticle : F Fc + F t F F F c m If is the angle made by [Note angle between and F t is 90º] F with, then tan Ft Fc tan 1 Ft Fc Angle between F & F t is (90º ) t (ii) Net acceleation towads the cente centipetal acceleation a c (iii) Net acceleation, a m ac a t The angle made by 'a' with a c, tan at ac Ft Fc F net m Special Note : (i) a c a t In both unifom & non-unifom cicula motion is pependicula to elocity ; so wok done by centipetal foce will be zeo in both the cases. (ii) In unifom cicula motion F t 0, as a t 0, so wok done will be zeo by tangential foce. Ex. But in non-unifom cicula motion F t 0, thus thee will be wok done by tangential foce in this case. ate of wok done by net foce in non-unifom cicula motion ate of wok done by tangential foce P dw Ft a c d x. Ft. Paticle s Cicula Motion with Vaiable Velocity A paticle of mass m is moing in a cicula path of constant adius such that its centipetal acceleation a c is aying with time t as a c k t, whee k is a constant. The powe delieed to the paticle by the foces acting on it will be - (A) mk t (B) mk t (C) m k t (D) mk t Sol.(D) Centipetal acceleation, a c Vaiable elocity k t k t k t The foce causing the elocity to aies d F m m k The powe delieed by the foce is, P F mk kt mk t Hence coect answe is (D) - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page - 11

12 - Download ou andoid App. Ex. elation between Centipetal & Tangential Acceleation in Cicula Motion A ca is moing in a cicula path of adius 100 m with elocity of 00 m/sec such that in each sec its elocity inceases by 100 m/s, the net acceleation of ca will be - (in m/sec) (A) (B) 10 7 (C) 10 (D) 100 Sol.(A) We know centipetal acceleation Ex.4 a c (tangential elocity ) adius (00) 100 a t 400 m/sec O Tangential acceleation a t 100 m/sec (gien) a c a net o ac a t acat cos90 ac a t ( 400) (100) m/s [emembe the angle between a t i.e. the tangential acceleation and a c i.e. the adial acceleation, is always 90º] Hence coect answe is (A) Non Unifom Cicula Motion The kinetic enegy of a paticle moing along a cicle of adius depends on distance coeed (s) as T as, whee a is constant. The foce acting on the paticle as a function of s will be - (A) as 1/ s as 1 (B) (C) as s (D) as Sol.(A) The kinetic enegy T as Note: 1 as as Centipetal foce o adial foce, Futhe as as... (1) a m d a m s... () ds a... () m Using () and () gies tangential acceleation, d a a t. m a a m s s m m a t as Tangential foce, F t ma t as As centipetal and tangential foce ae mutually pependicula, theefoe Total Foce, F F F c t as (as) as s 1 Hence coect answe is (A) In the aboe example the angle made by F fom the centipetal acceleation will be Fc tan F t F c F t as F c as / s - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page - 1

13 - Download ou andoid App. Motion in Vetical Cicle : Motion of a body suspended by sting : This is the best example of non-unifom cicula motion. When the body ises fom the bottom to the height h apat of its kinetic enegy conets into potential enegy Total mechanical enegy emains conseed Total (P.E. + K.E.) at A Total (P.E. + K.E.) at P mu h + 1 u gh g(1 cos ) u [Whee is length of the sting] Tension at a point P : (i) At point P equied centipetal foce (a) Net foce towads the cente : T cos, which poides equied centipetal foce. (b) Tangential foce fo the motion F t sin This foce etads the motion (ii) esults : (a) Tension at the lowest point A : T A A + (Hee 0º) T A mu + (b) Tension at point B : T B T B B mu 5 (c) Tension at point C : T C C ( 180º) mu T C (Hee 90º) Thus we conclude that T A > T C > T B and also T A T B 6 (iii) Cases : B A u C T A T C T C T B T cos T m [ g cos + ] m [u gl ( cos )] (a) If u > 5g In this case tension in the sting will not be zeo at any of the point, which implies that the paticle will continue the cicula motion. (b) If u 5g In this case the tension at the top most point (B) will be zeo, which implies that the paticle will just complete the cicula motion. - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page - 1

14 - Download ou andoid App. (c) Citical Velocity : The minimum elocity at (d) If which the cicula motion is possible The citical elocity at A The citical elocity at B The citical elocity at C 5g g g Also T A 6, T B 0, T C g < u < 5g In this case paticle will not follow cicula motion. Tension in sting becomes zeo somewhee between points C & B wheeas elocity emain positie. Paticle leaes cicula path and follow paabolic tajectoy (e) If u g In this case both elocity and tension in the sting becomes zeo between A and C and paticle will oscillate along semi-cicula path. (f) If u < g Ex.5 The elocity of paticle emains zeo between A and C but tension will not be zeo and the paticle will oscillate about the point A. Velocity at Minimum Point in Vetical Cicula Motion A paticle of mass m tied with a sting of length is eleased fom hoizontal as shown in fig. The elocity at the lowest potion will be - (A) (C) 1 g (B) g g (D) 1 g Sol.(B) Suppose be the elocity of paticle at the lowest position B. Ex.6 Accoding to conseation of enegy (K.E. + P.E.) at A (K.E. + P.E.) at B 0 + l g Hence coect answe is (B) Maximum Velocity in Vetical Cicula Motion A 4 kg balls is swing in a etical cicle at the end of a cod 1 m long. The maximum speed at which it can swing if the cod can sustain maximum tension of 16.6 N will be - (A) 6 m/s (C) 8 m/s Sol.(A) Maximum tension T Ex.7 o O B l (B) 6 m/s (D) 64 m/s T m/s Hence coect answe is (A) Tension at Minimum Point in Vetical Cicula Motion The sting of a pendulum is hoizontal. The mass of the bob is m. Now the sting is eleased. The tension in the sting in the lowest position is - (1) 1 () () (4) 4 Sol.(C) The situation is shown in fig. Let be the elocity of the bob at the lowest position. In this position the P.E. of bob is coneted into K.E. hence - l A - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page - 14

15 - Download ou andoid App. Sol.(B) The minimum speed at highest point of a etical cicle is gien by c g m/s Ex.8 l 1 gl...(1) If T be the tension in the sting, then T Fom (1) & () T Hence coect answe is (C)...() Citical Velocity at Minimum Point in Vetical Cicula Motion A ball is eleased fom height h as shown in fig. Which of the following condition hold good fo the paticle to complete the cicula path? 5 5 (A) h (B) h 5 5 (C) h < (D) h > Sol.(B) Accoding to law of conseation of enegy (K.E + P.E.) at A (K.E + P.E) at B Ex h gh But elocity at the lowest point of cicle, 5 5 g gh 5 g h Hence coect answe is (B) Citical Velocity at Maximum Point in Vetical Cicula Motion The oadway bidge oe a canal is the fom of an ac of a cicle of adius 0 m. What is the minimum speed with which a ca can coss the bidge without leaing contact with the gound at the highest point (g 9.8 m/s ) (A) 7 m/s (C) 89 m/s (B) 14 m/s (D) 5 m/s Ex.0 Hence coect answe is (B) Maximum Peiodic time in Vetical Cicula Motion A cane filled with wate is eoled in a etical cicle of adius 0.5 m and the wate does not fall down. The maximum peiod of eolution must be - (A) 1.45 (B).45 (C) (D) 4.5 Sol.(A) The speed at highest point must be Ex.1 > g, > g T T 0.5 T < < < < 1.4 sec g g 9. 8 Maximum peiod of eolution 1.4 sec Hence coect answe is (A) Vetical Semicicula Motion A paticle of mass m slides down fom the etex of semi-hemisphee, without any initial elocity. At what height fom hoizontal will the paticle leae the sphee- (A) (C) 8 5 (B) (D) 5 8 Sol.(A) Let the paticles leaes the sphee at height h, B h cos N When the paticle leaes the sphee i.e. N 0 A cos N - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page - 15

16 - Download ou andoid App. g cos...(1) Accoding to law of conseation of enegy (K.E. + P.E.) at A (K.E. + P.E.) at B h g ( h)...() Fom (1) & () h Also cos Hence coect answe is (A) Ex. Vetical Cicula Motion A body of mass m tied at the end of a sting of length l is pojected with elocity 4l g, at what height will it leae the cicula path - (A) 5 l (C) 1 l (B) 5 l (D) l Sol.(A) Let the body will hae the cicula path at height h aboe the bottom of cicle fom figue T + cos On leaing the cicula path T 0 9. BANKING OF TACKS When a ehicle moes ound a cue on the oad with sufficient speed, thee is a tendency of oe tuning fo the ehicle. To aoid this the oad is gien a slope ising outwads. The phenomenon is known as banking (i) Let thee be ehicle on a oad haing slope. nomal eaction of the gound Hoizontal component Vetical component sin cos It poides necessay centipetal foce sin It balances the weight of the ehicle cos g l cos...(1) Accoding to law of conseation of enegy (K.E. + P.E.) at A (K.E. + P.E.) at B 0 + l 1 + h g(l h)...() Fom (1) & () h 5 l h Also cos Hence coect answe is (A) tan g cos This equation gies the angle of banking equied. Conditions fo skidding and oetuning : Let thee be a ca moing on a oad moing on a cued path. a distance between the wheels h height of cente of gaitiy aboe the gound The foce acting on ca ae. (i) sin O Weight of ca W acting downwad (ii) Nomal eactions of gound a and b on inne and oute wheels espectiely (iii) The foce of fiction a and b Condition fo skidding : If is adius of cicula path, fo equilibium W a + b & a + b ( a + b ) This gies maximum speed fo skidding, max cos g B A - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page - 16

17 - Download ou andoid App. Condition fo oetuning : Taking moments about B, we get, a. a + h a 0 h a 1 ag If we take moments about A, we get b h 1 ag We know that b is always positie while a deceases as speed of the ca inceases. When h 1 ag a 0 i.e. inne wheel tends to loose contact with the eath. When h > 1 ag a Negatie i.e. the ca oetuns outwads. Thus the maximum speed fo no oetuing is gien by h 1 0 ag max Ex. Sol. ag h equied Centipetal Foce fo Motion on Cicula Path A ehicle of mass 1000 kg is moing along a cued both of length 14 m with a speed of 7 km/h. If it takes a tun of 90º, the centipetal foce needed by the ehicle is - (A) 0 N (C) 000 N (B) 00 N (D) N As the ehicle has a tun of 90º, the length 1 of the path is the pat of the cicle of 4 adius. Hence length of the path o m Centipetal foce, Ex.4 Hence coect answe is (C) N Necessay Condition fo Motion on Cicula Path Fo a heay ehicle moing on a cicula cue of a highway the oad bed is banked at an angle coesponding to a paticula speed. The coect angle of banking of the oad fo ehicles moing at 60 km/h will be - (If adius of cue 0.1 km) (A) tan 1 (0.8) (B) tan 1 (. 8) (C) tan 1 (0.05) (D) tan 1 (0.5) 50 Sol.(A) 60 km/h m/s 0.1 km 100m Ex.5 tan 0.8 g tan 1 (0.8) Hence coect answe is (A) A tain has to negotiate a cue of adius 400 m. By how much should the oute ail be aised with espect to inne ail fo a speed of 48 km/h. The distance between the ail is 1 m. (A) 1 m (B) 1 cm (C) 4.5 cm (D) 4.5 m Sol.(C) We know that tan... (1) g Let h be the elatie aising of oute ail with espect to inne ail. Then tan h... () (l sepaation between ails) Fom (1) & (), h g x l - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page - 17

18 - Download ou andoid App. Hence 48 km/h ( 400 m, l 1m), h 10 m/s, 9 (10 / 9) m 4.5 cm Hence coect answe is (C) POINTS TO EMEMBE 1. Centipetal foce does not incease the kinetic enegy of the paticle moing in cicula path, hence the wok done by the foce is zeo.. Centifuges ae the appaatuses used to sepaate small and big paticles fom a liquid.. The physical quantities which emain constant fo a paticle moing in cicula path ae speed, kinetic enegy and angula momentum. 4. If a body is moing on a cued oad with speed geate than the speed limit, the eaction at the inne wheel disappeas and it will leae the gound fist. 5. On unbanked cued oads the minimum adius of cuatue of the cue fo safe diing is /g, whee is the speed of the ehicle and is small. 6. If is the adius of cuatue of the speed beake, then the maximum speed with which the ehicle can un on it without leaing contact with the gound is (g) 7. W hile taking a tun on the leel oad sometimes ehicles o etun due to centifugal foce. 8. If h is the height of cente of gaity aboe the oad, a is half the wheel base then fo oad safety.h <. a, Minimum safe speed fo no oetuning is ( ga / h). the platfom, ( g / ), whee is the coefficient of fiction between the object and the platfom. 10. If an inclined plane ends into a cicula loop of adius, then the height fom which a body should slide fom the inclined plane in ode to complete the motion in cicula tack is h 5/. 11. Minimum elocity that should be impated to a pendulum to complete the etical cicle is ( 5g ), whee l is the length of the pendulum. 1. While descibing a etical cicle when the stone is in its lowest position, the tension in the sting is six times the weight of the stone. 1. The total enegy of the stone while eoling in etical cicle is (5/) l. 14. When the stone is in hoizontal position then the tension in the sting is and the elocity of the stone is ( g ). 15. If the elocity of the stone at the highest point is X, then the tension at the lowest point will be (X + 6). 16. If a body of mass m is tied to a sting of length l and is pojected with a hoizontal elocity u such that it does not complete the motion in the etical cicle, then (a) the height at which the elocity anishes is h u g (b) the height at which the tension anishes is h g g 17. K.E. of a body moing in hoizontal cicle is same thoughout the path but the K.E. of the body moing in etical cicle is diffeent at diffeent places. u 9. On a otating platfom, to aoid the skidding of an object placed at a distance fom axis of otation, the maximum angula elocity of - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page - 18

19 - Download ou andoid App. SOLVED EXAMPLES Ex.1 The magnitude of the linea acceleation, the paticle moing in a cicle of adius of 10 cm with unifom speed completing the cicle in 4 s, will be - (A) 5 cm/s (B).5 cm/s (C) 5 cm/s (D).5 cm/s Sol.(D) The distance coeed in completing the cicle is 10 cm The linea speed is Ex cm/s t 4 The linea acceleation is, (5) a.5 cm/s 10 This acceleation is diected towads the cente of the cicle Hence coect answe is (D) A cane filled with wate is eoled in a etical cicle of adius 4 m and wate just does not fall down. The time peiod of eolution will be - (A) 1 s (B) 10 s (C) 8 s (D) 4 s Sol.(D) We know that Ex. Time peiod Cicumfeence Citical speed sec g Hence coect answe is (D) The length of second's hand in a watch is 1 cm. The change in elocity of its tip in 15 seconds is - (A) 0 (B) (C) cm/s (D) 0 0 Sol.(B) Velocity Cicumfeence Time of eolution cm/s cm/s 0 60 cm/s Ex.4 Change in elocity Hence coect answe is (B) cm/s An electon is moing in a cicula obit of adius mete aound the atomic nucleus at a ate of eolutions pe second. The acceleation of the electon and centipetal foce acting on it will be - (The mass of the electon is kg) (A) N (C) N (B) N (D) N Sol.(A) Let the adius of the obit be and the numbe of eolutions pe second be n. Then the elocity of electon is gien by n, Ex.5 4 n Acceleation a 4 n Substituting the gien alues, we hae a 4 (.14) ( ) ( ) m/s towads the nucleus. The centipetal foce is F C ma ( ) ( ) N towads the nucleus. Hence coect answe is (A) An ai caft executes a hoizontal loop of adius 1 km with a steady speed of 900 km/h. The atio of centipetal acceleation to that gaitational acceleation will be- (A) 1 : 6.8 (B) 6. 8 : 1 (C).5 : 9.8 (D).5 : 9.8 Sol.(B) Gien that adius of hoizontal loop 1 km 1000 m Speed 900 km/h m/s Centipetal acceleation a c m/s - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page - 19

20 - Download ou andoid App. Ex.6 Centipetal acceleation Gaitational acceleation Hence coect answe is (B) a c g 6.8 : 1 A ca die is negotiating a cue of adius 100 m with a speed of 18 km/h. The angle though which he has to lean fom the etical will be - (A) tan (C) tan 1 Sol.(B) We know that, tan Ex.7 (B) tan (D) tan 1 0 g tan Hence coect answe is (B) Wite an expession fo the position ecto fo a paticle descibing unifom cicula motion, using ectangula co-odinates and the unit ectos i and j. The ecto expessions f o the elocity and acceleation a will be- (A) (B) / (C) (D) Sol.(D) î x + ĵ y, x cos, Ex.8 y sin whee t î ( cos t) + ĵ ( sin t) d/ î ( sin t) ĵ ( cos t) a d / Hence coect answe is (D) The etical section of a oad oe a canal bidge in the diection of its length is in the fom of cicle of adius 8.9 mete. Find the geatest speed at which the ca can coss this bidge without losing contact with the oad at its highest point, the cente of gaity of the ca being at a height h 1.1 mete fom the gound. (Take g 10 m/sec ) (A) 5 m/s (B) 7 m/s (C) 10 m/s (D) 1 m/s Sol.(C) Let be the nomal eaction exeted by the oad on the ca. At the highest point, we hae Ex.9, should not be negatie. ( h) Theefoe ( + a)g ( ) 10 o m/sec max 10 m/sec Hence coect answe is (C) The maximum speed at which a ca can tun ound a cue of 0 mete adius on a leel oad if the coefficient of fiction between the tyes and the oad is 0.4, will be - (A) m/s (B) m/s (C) m/s (D) 9.0 m/s Sol.(A) Let W Mg be the weight of the ca. Fiction foce 0.4 W Ex.10 Centipetal foce M 0.4 W W g W g 0.4 g m/sec Hence coect answe is (A) The angula speed with which the eath would hae to otate on it axis so that a peson on the equato would weight (/5) th as much as pesent will be: (Take the equatoial adius as 6400 km) (A) ad/sec (B) ad/sec (C) ad/sec (D) ad/sec Sol.(C) Let be the speed of eath's otation. We know that W Hence o W 5 5 o Now g 5 9.8( ) 5 - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page - 0

21 - Download ou andoid App. Ex.11 Soling, we get m/sec, g adian/sec. Hence coect answe is (C) A man whils a stone ound his head on the end of a sting 4.0 mete long. Can the sting be in a hoizontal, plane? If the stone has a mass of 0.4 kg and the sting will beak, if the tension in it exceeds 8 N. The smallest angle the sting can make with the hoizontal and the speed of the stone will espectiely be (Take g 10 m/sec ) (A) 0º, 7.7 m/s (B) 60º, 7.7 m/s (C) 45º, 8. m/s (D) 60º, 8.7 m/s Sol.(A) O A Tsin Fom figue T cos... (1) T sin l sin... () Fom eq. (1) T cos When the sting is hoizontal, must be 90º i.e.,cos 90º 0 T 0 Thus the tension must be infinite which is impossible, so the sting can not be in hoizontal plane. The maximum angle is gien by the beaking tension of the sting in the equation T cos m.g Hee T (Maximum) 8 N and m 0.4 kg 8 cos 0.4 g cos (4/8) 1, 60º The angle with hoizontal 90º 60º 0º T Tcos T Ex.1 Fom equation (), 8 sin 60º sin 60º sin 60º 80 sin 60º 7.7 m/sec Hence coect answe is (A) 0.4 o 4 sin 60 A smooth table is placed hoizontally and a sping of unsteched length l 0 and foce constant k has one end fixed to its cente. To the othe end of the sping is attached a mass m which is making n eolutions pe second aound the cente. Tension in the sping will be (A) 4 m k l 0 n / (k 4 m n ) (B) 4 m k l 0 n / (k + 4 m n ) (C) m k l 0 n / (k 4 m n ) (D) m k l 0 n / (k 4 m n ) Sol.(A) Let T be the tension poduced in the stetched sting. The centipetal f oce equied fo the mass m to moe in a cicle is poided by the tension T. The stetched length of the sping is (adius of the cicle). Now, Elongation poduced in the sping ( l 0 ) Tension poduced in the sping, T k ( l 0 )... (1) Whee k is the foce constant Linea elocity of the motion n Centipetal foce m(n) 4 n m... () Equating equation. (1) and (), we get k ( l 0 ) 4 n m ( T /) k k l 0 4 n m (k 4 n m) k l 0 k 0 (k 4 n m)...() Substituting the alue of in eqn. (1) we hae T k k 0 (k 4 n m) 0 - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page - 1

22 - Download ou andoid App. 4 n m0k o T (k 4 n m) Hence coect answe is (A)... (4) The ball now ises to a point D, whee its potential enegy is (h ). If D be the elocity of the ball at D, then, Ex.1 A moto ca is taelling at 0 m/s on a cicula oad of adius 500 m. It is inceasing its speed at the ate of m/s. Its net acceleation is (in m/s ) (A) (B) 1. 8 (C).7 (D) 0 Sol.(C) Two types of acceleation ae expeienced by the ca (i) adial acceleation due to cicula path, (0) a 1.8 m/s 500 (ii) A tangential acceleation due to incease of tangential speed gien by a t /t m/s adial and tangential acceleation ae pependicula to each othe. Net acceleation of ca a a a t 1.8) () Hence coect answe is (C) (.7 m/s Ex.14 In figue ABCDE is a channel in the etical plane, pat BCDE being cicula with adius. A ball is eleased fom A and slides without fiction and without olling. It will complete the loop path - (A) if h is geate than 5/ (B) if h is less than 5/ (C) if h is geate than /5 (D) if h is less than /5 Sol.(A) h A Let m be the mass of the ball. When the ball comes down to B, its potential enegy h which is coneted into kinetic enegy. Let B, be the elocity of the ball at B. Then, h 1 m B E D B C m g (h ) 1 m D...() Now to complete the cicula path, it is necessay that the centifugal foce acting upwad at point D, should be equal o geate than the foce acting downwad at point D should be equal o geate than the foce acting downwad. Theefoe D o D g Fom equation () D g (h ), g (h ) g h 5 Hence coect answe is (A) Ex.15 An aicaft loops the loop of adius 500 m with a constant elocity 60 km/hou. The weight of the flye of mass m 70 kg in the lowe, uppe and middle points of the loop will espectiely be- (A) 10 N, 700 N, 1400 N (B) 1400 N, 700 N, 100 N (C) 700 N, 1400 N, 10 N, (D) 100 N, 700 N, 1400 N Sol.(D) See fig, Hee 60 km/h 100 m/sec At lowe point, N, N weight of the flye + N N N N 70(10000) N At uppe point, N +, - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page -

23 - Download ou andoid App. Ex.16 N N At middle point, N 1400 N Hence coect answe is (D) A paticle of mass kg is moing unde the action of a cental foce whose potential enegy is gien by U() 10 joule. Fo what enegy and angula momentum will the obit be a cicle of adius 10 m- (A) J, 000 kgm /sec (B) J, 000 kgm /sec (C).5 10 J, 00 kgm /sec (D).5 10 J, 00 kgm /sec Sol.(A) Gien that U() 10 So the foce F acting on the paticle is gien by, U F (10 ) 10 0 Fo cicula motion of the paticle, F 0 Substituting the gien alues, we hae, 10 0 (10) o 100 m/s The total enegy in cicula motion E K.E. + P.E. 1 + U() 1 (100) (10) joule Angula momentum kg m /sec Also time peiod T sec Hence coect answe is (A) - Fee NEET & IIT Study Meteial & Papes - Download ou andoid App. Page -