ISSUED BY K V - DOWNLOADED FROM CIRCULAR MOTION

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1 K.V. Silcha CIRCULAR MOTION Cicula Motion When a body moves such that it always emains at a fixed distance fom a fixed point then its motion is said to be cicula motion. The fixed distance is called the adius of the cicula path and the fixed point is called the cente of the cicula path. Unifom Cicula Motion Cicula motion pefomed with a constant speed is known as unifom cicula motion. Angula Displacement Angle swept by the adius vecto of a paticle moving on a cicula path is known as angula displacement of the paticle. Example : angula displacement of the paticle fom P 1 to P is. P P 1 Relation Between Angula Displacement and Linea Displacement Since, Angle = ac adius Anglula Displacement = ac P 1 P adius = s Angula Velocity Rate of change of angula displacement of a body with espect to time is known as angula displacement. It is epesented by ω. Aveage Angula Velocity It is defined as the atio of total angula displacement to total time taken. ω avg = Total Angula Displacement Total Time Taken ω avg = t Instantaneous Angula Velocity Angula velocity of a body at some paticula instant of time is known as instantaneous angula velocity. O 61

2 K.V. Silcha Aveage angula velocity evaluated fo vey shot duation of time is known as instantaneous angula velocity. ω = Lim ω avg = t 0 t ω = d dt Relation Between Angula Velocity and Linea Velocity We know that angula velocity Putting, = s/ o, o, o, ω = d dt ω = d (s/) dt ω = 1 ds dt ω = v v = ω Time Peiod of Unifom Cicula Motion Total time taken by the paticle pefoming unifom cicula motion to complete one full cicula path is known as time peiod. In one time peiod total angle otated by the paticle is and time peiod is T. Hence angula velocity ω = T o, T = ω Fequency - Numbe of evolutions made by the paticle moving on cicula path in one second is known as fequency. f = 1 = ωa T Centipetal Acceleation When a body pefoms unifom cicula motion its speed emains constant but velocity continuously changes due to change of diection. Hence a body is continuously acceleated and the acceleation expeienced by the body is known as centipetal acceleation (that is the acceleation diected towads the cente). 6

3 K.V. Silcha v a P v 1 v v O s C B R v v 1 v v P 1 A Conside a paticle pefoming unifom cicula motion with speed v. When the paticle changes its position fom P 1 to P its velocity changes fom v 1 to v due to change of diection. The change in velocity fom P 1 to P is v which is diected towads the cente of the cicula path accoding to tiangle law of subtaction of vectos. Fom figue OP 1 P and ABC ae simila, hence applying the condition of similaity BC = P 1 P 1 AB O P 1 o, v = s v o, v = v s Dividing both sides by t, v = v s t t Taking limit t 0 both sides, Lim v = v Lim t 0 t t 0 t o, dv = vds dt dt o, a = v Putting v = ω, a = ω Since the change of velocity is diected towads the cente of the cicula path, the acceleation esponsible fo the change in velocity is also diected towads cente of cicula path and hence it is known as centipetal acceleation. Centipetal Foce Foce esponsible fo poducing centipetal acceleation is known as centipetal foce. Since centipetal acceleation is diected towads the cente of the cicula path the centipetal foce is also diected towads the cente of the cicula path. If a body is pefoming unifom cicula motion with speed v and angula velocity ω on a cicula path of adius, then centipetal acceleation is given by 63

4 K.V. Silcha F c F c = mv = mω Net Acceleation of a Body Pefoming Non-Unifom Cicula Motion When a body pefoms non-unifom cicula motion its speed i.e. magnitude of instantaneous velocity vaies with time due to which it expeiences tangential acceleation a T along with the centipetal acceleation a C. Since both the acceleations act simultaneously on a body and ae mutually pependicula to each othe, the esultant acceleation a R is given by thei vecto sum. a R a T a C a R = a T + a C a R = a T + a C Physical Application of Centipetal Foce i) Case - 1 Cicula motion of a stone tied to a sting. Centipetal foce is povided by the tension of the sting F c = mv = T ii) Case - Cicula motion of electon aound the nucleus. Centipetal foce is povided by the electostatic foce of attaction between the positively chaged nucleus and negatively chaged electon F c = mv = F E 64

5 K.V. Silcha iii) Case - 3 Cicula motion of planets aound sun o satellites aound planet. Centipetal foce is povided by the gavitational foce of attaction between the planet and sun F c = mv = F g iv) Case - 4 Cicula motion of vehicles on a hoizontal oad. Centipetal foce is povided by the static fictional foce between the oad and the tye of the vehicle. F c = mv = f s v) Case - 5 Cicula motion of a block on otating platfom. Centipetal foce is povided by the static fictional foce between the block and the platfom. F c = mv = f s vi) Case - 6 Cicula motion of mud paticles sticking to the wheels of the vehicle. Centipetal foce is povided by the adhesive foce of attaction between the mud paticles and the tyes of the vehicle. F c = mv = F adhesive At vey high speed when adhesive foce is unable to povide necessay centipetal foce, the mud paticles fly off tangentially. In ode to pevent the paticles fom staining ou clothes, mud-guads ae povided ove the wheels of vehicle. v v 65

6 K.V. Silcha vii) Case - 7 Cicula motion of a tain on a hoizontal tack. Centipetal foce is povided by the hoizontal component of the eaction foce applied by the oute tack on the inne pojection of the oute wheels N N H F c = mv = N Hoizontal N H viii) Case - 8 Cicula motion of a toy hanging fom ceiling of vehicle. T TCos TSin Ca moving with constant velocity on hoizontal oad Ca taking a tun with constant velocity on a hoizontal oad Wheneve ca takes a tun, sting holding the toy gets tilted outwad such that the vetical component of the tension of sting balances the weight of the body and the hoizontal component of tension povides the necessay centipetal foce. TSin = mv TCos = Theefoe, Tan = v ix) Case - 9 Conical pendulum. g T TSin T a TCos Wheneve bob of a pendulum moves on a hoizontal cicle it s sting geneates a cone. Such a pendulum is known as conical pendulum. The vetical component of the tension of the sting balances the weight of the body and the hoizontal component of tension povides the necessay centipetal foce. 66

7 K.V. Silcha TSin = mv TCos = Theefoe, Tan = v x) Case - 10 Well of death. g N NCos NCos In the well of death, the ide ties to push the wall due to its tangential velocity in the outwad diection due to which wall applies nomal eaction in the inwad diection. The vetical component of the nomal eaction balances the weight of the body and its hoizontal component povides the necessay centipetal foce. NSin = mv NCos = Theefoe, Tan = v g xi) Case - 11 Tuning of aeo plane. F P F P Cos F P a F P Sin While taking a tun aeo-plane tilts slightly inwads due to which it s pessue foce also gets tilted inwads due to which it s pessue foce also gets tilted inwads such that it s vetical component balances the weight of the body and the hoizontal component povides the necessay centipetal foce. 67

8 K.V. Silcha F P Sin = mv F P Cos = Theefoe, Tan = v xi) Case - 11 Banking of Roads In case of hoizontal oad necessay centipetal foce mv / is povided by static fictional foce. When heavy vehicles move with high speed on a shap tun (small adius) then all the factos contibute to huge centipetal foce which if povided by the static fictional foce may esult in the fatal accident. To pevent this oads ae banked by lifting thei oute edge. Due to this, nomal eaction of oad on the vehicle gets tilted inwads such that it s vetical component balances the weight of the body and the hoizontal component povides the necessay centipetal foce. g n ncosa c nsin nsin = mv ncos = 68

9 K.V. Silcha Theefoe, Tan = v g xii) Case - 1 Bending of Cyclist In case of a cyclist moving on a hoizontal cicula tack necessay centipetal foce is povided by static fictional foce acting paallel along the base. As this fictional foce is not passing fom the cente of mass of the system it tends to otate the cycle along with the cyclist and make it fall outwad of the cente of the cicula path. To pevent himself fom falling, the cyclist leans the cycle inwads towads the cente of the cicle due to which the nomal eaction of the suface of oad on the cycle also leans inwad such that that its vetical component balances the weight of the body and the hoizontal component povides the necessay centipetal foce. N NCos NSin NSin = mv NCos = Theefoe, Tan = v g 69

10 K.V. Silcha xiii) Case - 13 Motion of a Ball in a Bowl ω A o a N NCos NCos When the bowl otates with some angula velocity ω. The vetical component of the nomal eaction of the bowl on the ball balances the weight of the body and its hoizontal component povides the necessay centipetal foce. NSin = mv NCos = Theefoe, Tan = v xiv) Case - 14 Motion of a tain on the banked tacks. At the tuns tacks ae banked by slightly elevating the oute tacks with espect to the inne ones. This slightly tilts the tain inwads towads the cente of the cicula path due to which the nomal eaction of the tacks on the tain also gets slightly tilted inwads such that the vetical component of the nomal eaction balances the weight of the tain and it s hoizontal component povides the necessay centipetal foce. g N NCos 70

11 K.V. Silcha NSin = mv NCos = Theefoe, Tan = v g Vetical Cicula Motion Wheneve the plane of cicula path of body is vetical its motion is said to be vetical cicula motion. Vetical Cicula Motion of a Body Tied to a Sting v A A T A T A Conside a body of mass m tied to a sting and pefoming vetical cicula motion on a cicula path of adius. At the topmost point A of the body weight of the body and tension T A both ae acting in the vetically downwad diection towads the cente of the cicula path and they togethe povide centipetal foce. T A + = mv A Citical velocity at the top most point As we go on deceasing the v A, tension T A also goes on deceasing and in the citical condition when v A is minimum tension T A = 0. The minimum value of v A in this case is known as citical velocity T A(Citical) at the point A. Fom above 0 + = mv A(Citical) o, o, v A(Citical) = g v A(Citical) = g 71

12 K.V. Silcha If the velocity at point A is less than this citical velocity then the sting will slag and the body in spite of moving on a cicula path will tend to fall unde gavity. Citical velocity at the lowe most point A T B T B V B B Taking B as efeence level of gavitational potential enegy and applying enegy consevation E A = E B P A + K A = P B + K B + 1mv A = 0 + 1mv B Putting, v A = g + 1m( g) = 0 + 1mv B o, 4 + = mv B o, 5 = mv B o, v B = 5g This is the minimum possible velocity at the lowe most point fo vetical cicula motion known as citical velocity at point B. v B(Citical) = 5g Tension at lowemost point in citical condition Fo lowemost point B net foce towads the cente is centipetal foce. Tension T B acts towads the cente of the cicula path wheeas weight acts away fom it. Hence, T B = mv B Putting, v B = 5g 7

13 K.V. Silcha T B = m5g o, T B = 6 Hence in citical condition of vetical cicula motion of a body tied to a sting velocities at topmost and lowemost be (g) and (5g) espectively and tensions in the stings be 0 and 6 espectively. Geneal Condition fo Slagging of Sting in Vetical Cicula Motion Fo the body pefoming vetical cicula motion tied to a sting, slagging of sting occus in the uppe half of the vetical cicle. If at any instant sting makes angle with vetical then applying net foce towads cente is equal to centipetal foce, we have Cos T v Fo slagging T = 0, o, T + Cos = mv 0 + Cos = mv v = gcos Case-1 At Topmost point = 0, theefoe v = g Case- At = 60 o, theefoe v = gcos60 = g/ Case-3 When sting becomes hoizontal that is at = 90 o, v = gcos90 = 0 Velocity Relationship at diffeent Points of Vetical Cicula Motion Let initial and final velocities of the body pefoming vetical cicula motion be v 1 and v and the angle made by sting with the vetical be 1 and. Taking lowemost point of vetical cicula path as efeence level and applying enegy consevation, 73

14 K.V. Silcha V 1 V Cos 1 Cos 1 E 1 = E P 1 + K 1 = P + K ( + Cos 1 ) + 1mv 1 = ( + Cos ) + 1mv o, (Cos 1 Cos ) = 1m(v v 1 ) o, (v v 1 ) = g(cos 1 Cos ) Vetical Cicula Motion of a Body Attached to a Rod Since od can neve slag hence in the citical situation a body attached to the od may each the topmost position A of the vetical cicula path with almost zeo velocity. In this case its weight acts in vetically downwad diection and tension of od acts on the body in the vetically upwad diection. Applying net foce towads cente is equal to centipetal foce, v A A T A T A - T A = mv A Putting v A = 0 (fo citical condition) - T A = 0 o, T A = 74

15 K.V. Silcha Citical velocity and Tension at the lowe most point A T B T B V B B Taking B as efeence level of gavitational potential enegy and applying enegy consevation E A = E B P A + K A = P B + K B + 1mv A = 0 + 1mv B Putting, v A = 0(fo citical condition) + 0 = 0 + 1mv B o, 4 = mv B o, v B = 4g This is the minimum possible velocity at the lowe most point fo vetical cicula motion known as citical velocity at point B. v B(Citical) = 4g Tension at lowemost point in citical condition Fo lowemost point B applying net foce towads cente is equal to centipetal foce. Tension T B acts towads the cente of the cicula path wheeas weight acts away fom it in vetically downwad diection. Hence, Putting, v B = 4g o, T B = mv B T B = m4g T B = 5 75

16 K.V. Silcha Hence in citical condition of vetical cicula motion of a body attached to the od velocities at topmost and lowemost be 0 and 4g espectively and tensions in the od be (pushing natue) and 5 (pulling natue) espectively. Motion of A Body Ove Spheical Suface N V Sin Cos A body of mass m is moving ove the suface of the smooth sphee of adius. At any instant when the adius of sphee passing though the body makes angle with the vetical the tangential velocity of the body is v. Since net foce towads the cente is centipetal foce we have Cos N = mv o, N = Cos mv if v inceases N deceases and when the body just loses contact with the sphee N = 0. Putting N = 0, o, o, 0 = Cos mv mv = Cos v = g Cos This is the minimum velocity at which the body loses contact and it is the maximum velocity at which the body emains in contact with the suface. 76

17 K.V. Silcha CENTRIFUGAL FORCE It is a pseudo foce expeienced by a body which is a pat of the cicula motion. It is a non-ealistic foce and comes into action only when the body is in a cicula motion. Once the cicula motion of the body stops, this foce ceases to act. Its magnitude is exactly same as that of centipetal foce but it acts opposite to the diection of the centipetal foce that is in the adially outwad diection. Fame of efeence attached to a body moving on a cicula path is a non-inetial fame since it an acceleated fame. So when eve any body is obseved fom this fame a pseudo foce F = ma = mv / = mω must be applied on the body opposite to the diection of acceleation along with the othe foces. Since the acceleation of the fame in cicula motion is centipetal acceleation a = v / diected towads the cente of the cicula path, the pseudo foce applied on the bodies obseved fom this fame is F = mv / diected away fom the cente of the cicula path. This pseudo foce is temed as a centifugal foce. F CF F Centifugal = mv = mω (Diected in adially outwad diection) CENTRIFUGE It is an appaatus used to sepaate ceam fom milk. It woks on the pincipal of centifugal foce. It is a cylindical vessel otating with high angula velocity about its cental axis. When this vessel contains milk and otates with high angula velocity all the paticles of milk stat moving with the same angula velocity and stat expeiencing centifugal foce F Centifugal = mω in adially outwad diection. Since centifugal foce is diectly popotional to the mass of the paticles, massive paticles of milk on expeiencing geate centifugal foce stats depositing on the oute edge of the vessel and lighte ceam paticles on expeiencing smalle centifugal foce ae collected nea the axis fom whee they ae sepaated apat. ω Ceam 77

18 K.V. Silcha MEMORY MAP ω = v = = f T Centipetal Foce F C = mv = mω Radially Outwad Diection Cicula Motion Centifugal Foce F CF = mv = mω Radially Inwad Diection Body Tied to Sting V top = (g) and V bottom = (5g) T top = 0 and T bottom = 6 Citical Condition Fo Vetical Cicula Motion Body Attached to Rod V top = 0 and V bottom = (4g) T top = - and T bottom = 5 Citical Condition of Vetical Cicula MOtion 78

19 K.V. Silcha Vey Shot Answe Type 1 Mak Questions 1. Is net foce needed to keep a body moving with unifom velocity?. Is Newton s nd law (F = ma) always valid. Give an example in suppot of you answe? 3. Action and eaction foces do not balance each othe. Why? 4. Can a body emain in state of est if moe than one foce is acting upon it? 5. Is the centipetal foce acting on a body pefoming unifom cicula motion always constant? 6. The sting is holding the maximum possible weight that it could withstand. What will happen to the sting if the body suspended by it stats moving on a hoizontal cicula path and the sting stats geneating a cone? 7. What is the eaction foce of the weight of a book placed on the table? 8. What is the maximum acceleation of a vehicle on the hoizontal oad? Given that coefficient of static fiction between the oad and the tyes of the vehicle is µ. 9. Why guns ae povided with the shoulde suppot? 10. While paddling a bicycle what ae the types of fiction acting on ea wheels and in which diection? Answe 1. No.. It is valid in an inetial fame of efeence. In non-inetial fame of efeence (such as a ca moving along a cicula path), Newton s nd law doesn t hold appaently. 3. Since they ae acting on diffeent bodies. 4. Yes, if all the foces acting on it ae in equilibium. 5. No, only its magnitude emains constant but its diection continuously goes on changing. 6. It will beak because tension in the sting inceases as soon as the body stats moving. 7. The foce with which the book attacts the eath towads it. 8. a max = fs(max)/m = µn/m = µ/m = µg. 9. So that the ecoil of gun may be educed by poviding suppot to the gun by the shouldes. 10. Static fiction in fowad diection and olling fiction in backwad diection. Shot Answe Type Maks Questions 1. Explain why the wate doesn t fall even at the top of the cicle when the bucket full of wate is upside down otating in a vetical cicle?. The displacement of a paticle of mass 1kg is descibed by s = t + 3t. Find the foce acting on paticle? (F = 6N) 3. A paticle of mass 0.3 kg is subjected to a foce of F = -kx with k = 15 Nm 1. What will be its initial acceleation if it is eleased fom a point 10 cm away fom the oigin? (a = - 5 ms ) 4. Thee foces F 1, F and F 3 ae acting on the paticle of mass m which is stationay. If F 1 is emoved, what will be the acceleation of paticle? (a = F 1 /m) 79

20 K.V. Silcha 5. A sping balance is attached to the ceiling of a lift. When the lift is at est sping balance eads 50 kg of a body hanging on it. What will be the eading of the balance if the lift moves :- (i) Vetically downwad with an acceleation of 5 ms (ii) Vetically upwad with an acceleation of 5 ms (iii) Vetically upwad with a constant velocity. Take g = 10m/s. [(i) 5kgf,(ii) 75kgf, (iii) 50kgf] 6. Is lage suface aea beak on a bicycle wheel moe effective than small suface aea bake? Explain? 7. Calculate the impulse necessay to stop a 1500 kg ca moving at a speed of 5ms 1? ( N-s) 8.Give the magnitude and diections of the net foce acting on a ain dop falling feely with a constant speed of 5 m/s? (F net = 0) 9. A block of mass.5kg ests on a smooth hoizontal table. What steady foce is equied to give the block a velocity of m/s in 4 s? (F=.5N) 10. Calculate the foce equied to move a tain of 00 quintal up on an incline plane of 1 in 50 with an acceleation of ms. The foce of fiction pe quintal is 0.5 N? (F = 44100N) Shot Answe Type 3 Maks Questions 1. A bullet of mass 0.0 kg is moving with a speed of 10m 1 s. It penetates 10 cm of a wooden block befoe coming to est. If the thickness of the taget is educed to 6 cm only find the KE of the bullet when it comes out? (Ans : 0.4 J). A man pulls a lawn olle with a foce of F. If he applies the foce at some angle with the gound. Find the minimum foce equied to pull the olle if coefficient of static fiction between the gound and the olle is µ? 3. A ball bounces to 80% of its oiginal height. Calculate the change in momentum? 4. A pendulum bob of mass 0.1 kg is suspended by a sting of 1 m long. The bob is displaced so that the sting becomes hoizontal and eleased. Find its kinetic enegy when the sting makes an angle of (i) 0, (ii) 30, (iii) 60 with the vetical? 5. The velocity of a paticle moving along a cicle of adius R depends on the distance coveed s as F = αs whee α is constant. Find the foce acting on the paticle as a function of s? 6. A block is pojected hoizontally on ough hoizontal floo with initial velocity u. The coefficient of kinetic fiction between the block and the floo is µ. Find the distance tavelled by the body befoe coming to est? 7. A locomotive of mass m stats moving so that its velocity v changes accoding to v = (α s), whee α is constant and s is distance coveed. Find the foce acting on the body afte time t? 8. Deive an expession fo the centipetal foce? 9. Find the maximum value of angle of fiction and pove that it is equal to the angle of epose? 10. State and pove Lami s theoem? Long Answe Type 5 Maks Questions 80

21 K.V. Silcha 1. Find the maximum and minimum velocity of a vehicle of mass m on a banked oad of banking angle, if coefficient of static fiction of the wheels of vehicle with the oad is µ?. Find the maximum and minimum foce applied paallel up the incline on a block of mass m placed on it if angle of inclination is and coefficient of static fiction with the block is µ so that the block emains at est? 3. Pove that in case of vetical cicula motion cicula motion of a body tied to a sting velocities at topmost and lowemost point be (g) and (5g) espectively and tensions in the stings be 0 and 6 espectively? 4. Find the maximum hoizontal velocity that must be impated to a body placed on the top of a smooth sphee of adius so that it may not loose contact? If the same body is impated half the velocity obtained in the fist pat then find the angula displacement of the body ove the smooth sphee when it just loses contact with it? 5. Find the acceleation of the blocks and the tension in the stings? m 3 m m 1 F Some Intellectual Stuff 1. Find the acceleation of the blocks m 1 and m. All the sufaces ae smooth and sting and pulley ae light? Also find the net foce on the clamped pulley? t m 1 m x. A body of mass m explodes into thee fagments of with masses in the atio ::6. If the two simila masses move of pependicula to each othe with the speed of 10m/s each, find the velocity of the thid paticle and its diection elative to the two othe bodies? 3. A mass of 5 kg is suspended by a ope of length m fom the ceiling. A hoizontal foce of 50 N is applied at the mid point P of the ope? Calculate the angle that the ope makes with the vetical and the tension in the pat of the ope between the point of suspension and point P?. Neglect the mass of the ope. (g = 10ms ) 4. A body moving inside a smooth vetical cicula tack is impated a velocity of (4g) at the lowemost point. Find its position whee it just loses contact with the tack? 5. m m 1 m m 1 81

22 K.V. Silcha Find in both the cases (i)acceleation of the two blocks.(ii)tension in the clamp holding the fixed pulley? 6. Mass of both the blocks is m and coefficient of kinetic fiction with the gound is µ. Find the acceleation of the two blocks and tension in the sting attached between the two blocks? F 7. A small sphee of mass m is placed in a hemispheical bowl of adius R. Bowl is otated with angula velocity ω. Find the angle made by the adius of the bowl passing though the sphee with the vetical when the sphee stats otating with the bowl? ω o R a 8. Mass of both the blocks is m find net foce on the pulley? 9. Mass of both the blocks is m find acceleation of both the blocks and net foce on the clamp holding the fixed pulley? 10. Mass of both the blocks is m find acceleation of the system and the tension in the od? F 8