Shree Datta Coaching Classes, Contact No Circular Motion

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Shee Datta Coaching Classes, Contact No. 93698036 Pof. Deepak Jawale Cicula Motion Definition : The motion of the paticle along the cicumfeence of a cicle is called as cicula motion. Eg.

1 Shee Datta Coaching Classes, Contact No Pof. Deepak Jawale Cicula Motion Definition : The motion of the paticle along the cicumfeence of a cicle is called as cicula motion. Eg. i) Motion of eath aound the sun. ii) Motion of moon o satellite aound the eath. iii) Motion of electon aound the nucleous. i) Motion of tip of second hand, minute hand and hou hand is a cicula motion. Radius ecto : The line joining the cente of a cicle and the paticle pefoming cicula motion epesents the adius ecto at that point. It has a constant magnitude but aiable diection Angula displacement ( o ) : The angle subtended by the adius ecto at the cente of a cicle in a gien time is called as angula displacement. [Angula displacement] [M 0 L 0 T 0 ] S.I. unit of angula displacement is adian (ad) Infinitesimal angula displacement is a ecto quantity. Angula elocity () : The ate of change of angula displacement with espect to time is called as angula elocity. d OR lim dt t 0 t S.I. unit of angula elocity is ad/s [Μ L Τ ] [M 0 L 0 T ] [T ] Angula elocity is a ecto quantity. Angula acceleation () : The ate of change of angula elocity with espect to time is called as angula acceleation. t 0 0 [Μ L Τ ] [M 0 L 0 T ] [T ] S.I. unit of angula acceleation is ad/s It is a ecto quantity. Right Hand Rule : Imagine the axis of otation to be held in ight hand with finges culed ound the axis. If the culed finges denote the sense of otation then the thumb denotes the diection of angula acceleation o angula elocity o infinitesimal angula displacement. δθ Cicula Motion Page

2 Shee Datta Coaching Classes, Contact No Pof. Deepak Jawale Q. Show that Ans. Conside a paticle pefoming cicula motion with linea elocity and angula elocity. Suppose that paticles moes fom pt. A to pt. B in infinitesimal small time t. As t 0 Then 0 Ac AB Chod AB i.e. the displacement of the paticle is almost along the staight line. Now, elocity displacement time But s lim t 0. lim t 0 lim. t t t 0 s t But lim t 0 t In ecto fom x Q. Show that a Solution : Conside a paticle pefoming cicula motion with linea elocity and angula elocity. Fo cicula motion, we hae Diffeentiate this equation with espect to time d d d d ( ) a dt dt dt dt d dt d dt a Unifom cicula motion (UCM) : The motion of a paticle along the cicumfeence of a cicle with a constant speed is called as unifom cicula motion. Unifom cicula motion is an acceleated motion. Since the diection of elocity of paticle which is always along the tangent dawn to the cicula path is continuously goes on changing duing the motion hence UCM is an acceleated motion. O Peiod of UCM : The time taken by a paticle to complete one eolution pefoming UCM is called as its peiod. Dist. Coeed in one eolution Cicumfeence Now, Peiod Velocity Velocity Cicula Motion Page

3 Shee Datta Coaching Classes, Contact No T But Pof. Deepak Jawale T T Fequency of UCM The numbe of eolutions made by a paticle pefoming UCM pe unit time is called as its fequency. n T But T n / n OR n Radial acceleation of centipetal acceleation : The acceleation of a paticle pefoming UCM is along the adius and is diected towads the cente is called as adial acceleation. / O Q.3 Obtain an expession fo adial acceleation of a paticle pefoming UCM. Ans. Solution : Conside a paticle pefoming UCM with linea elocity and angula elocity. Suppose that the paticle moes fom A to B in infinitesimal small time t. Let AP and BQ epesents elocity of paticle at pt-a and B espectiely. Daw BR equal and paallel to AP and join RQ. Since paticle pefoming UCM hence it speed is constant i.e. AP BQ By law of tiangle ectos BQ BR + RQ RQ BQ BR RQ BQ AP Velocity of paticle at pt. B elocity of paticle at pt.a Change in elocity RQ Now AOB RBQ S-A-S test RQ AB BQ OB Also, RQ AB OB x BQ AB t OB and BQ t RQ x Cicula Motion Page 3

4 Shee Datta Coaching Classes, Contact No t Pof. Deepak Jawale t Now, Acceleation lim t 0 t lim t 0 /. t t t lim t t 0 Radial acceleation The magnitude of acceleation is thus / o. The diection of acceleation is same as that of ecto RQ. If the time inteal t is continuously educe then the point B will appoach the point A i.e. As t 0 Then 0. Theefoe, RQ will be pependicula to the tangent AP will be along the adius and diected towads the cente. Hence it is adial acceleation. Centipetal Foce : In unifom cicula motion the foce acting on a paticle is along the adius and is diected towads the cente is called as centipetal foce. Explanation :- ) This foce is necessay fo unifom cicula motion. ) Centipetal foce mass x centipetal acceleation i.e. F m / m 3) It is a eal foce since it aises due to known inteactions. Example : ) If an object tied to a sting is eoled unifomly in a hoizontal cicle, the centipetal foce is due to the tension impated to the sting by the hand. ) The cicula motion of the moon ound the eath is due the gaitational attaction between the eath and the moon. In this case, the gaitational foce exeted by the eath on the moon acts as the centipetal foce. 3) In an atom, an electon eoles in a cicula obit ound the nucleus. In this case, the electostatic foce of attaction exeted by the positiely chaged nucleus on the negatiely chaged electon acts as the centipetal foce. Centifugal foce : In an acceleated fame of efeence, the pseudo foce expeienced by the paticle pefoming UCM is along the adius and is diected away fom the cente is called as centifugal foce. Explanation :- ) It is a pseudo foce since it is not aises due to any known inteaction. ) Centifugal foce mass x centipetal acceleation. i.e. F m / m. Example : ) When a ehicle moes along a cued path, a passenge in the ehicle expeiences a push in the outwad diection. This is due to the centifugal foce acting on the passenge in a diection along the adius away fom the cente of the cue. ) Place a coin on a gamophone disc nea its cente and set the disc into otation. As the speed of otation inceases, the coin will slide away fom the cente of the disc. The motion of the coin is due to the centifugal foce acting on it. Q.4 Distinguish between Centipetal and Centifugal foce. Centipetal foce Centifugal foce ) This foce is necessay fo the cicula motion. ) The imaginay foce expeienced by the body pefoming cicula motion. ) It is eal foce. ) It is pseudo foce. 3) It is diected adially towads the 3) It is diected adially away fom the cente. cente. 4) It exists in the inetial fame of efeence. 4) It is pesent in the non-inetial fame of efeence. Cicula Motion Page 4

5 Shee Datta Coaching Classes, Contact No Pof. Deepak Jawale Q.5 Obtain an expession fo safety speed fo a ehicle moing along a plane cue oad. Ans. Conside a ehicle of mass m moing along a cue hoizontal oad with elocity. The necessay centipetal foce is supplied by the fiction between the wheels and the suface of the oad. If is the coefficient of fiction between the wheels and the oad suface, the maximum foce of fiction is mg. Fo equilibium of the ca, Centipetal foce foce of fiction m mg g g This is the maximum speed with which a ehicle can be safely dien along the cue on a hoizontal oad. Banking of oad : The suface of the oad along the cue is kept inclined to the hoizontal at some suitable angle. So that the oute edge of the oad is at a highe eleation than the inne edge. This aangement of oad is called as banking of oad. Q.6 Explain the necessity of banking of oad. Ans : Safety speed fo a ehicle moing along a plane cue oad is gien by g This is the maximum speed with which ehicle can coss safely along cue hoizontal oad. If speed of ehicle is aboe this limit then ehicle may be in dange being thown away fom the oad. One way of potecting the ehicle moing at high speed fom accidents along cue oad is to use foce of fiction by making oad suface moe ough. But it inceases wea and tea of tyes of ehicle. Also foce of fiction is not eliable because it deceases when oil fom othe ehicles fall on the oad o when it wets due to ain. So the best and safe way of potecting a ehicle moing along at high speed fom accident along cue oad it to use banking of oad. In banked cue oad the hoizontal component of nomal eaction N sin poides necessay centipetal foce equied fo cicula motion. Q.7 Explain why Centifugal foce is called a Pseudo foce? Ans. The foces whose oigin is not due to known inteactions like gaitational, electomagnetic o nuclea ae called as pseudo foces. These foces ae pesent only in acceleated fame of efeence and they anish if the acceleation of the fame of efeence is made zeo. The centifugal is a pseudo foce as it is not poduced due to known inteactions in natue. It is the pseudo foce imagined in cicula motion so that Newton s laws of motion can be applied in the acceleated fame of efeence. Q.8 Obtain an expession fo safety speed fo ehicle moing along a banked cue oad. OR Show that angle of banking is independent of mass of ehicle. Ans. Conside a ehicle of mass m moing along a banked cue oad of adius and banked at an angle. Duing the cic7ula motion of a ehicle the foces ae acting on the ehicle a) its weight m g, acting etically downwad. b) the nomal eaction N of the oad, pependicula to the oad suface. c) the fictional foce fs along the If s is the coefficient of static fiction between the tyes and oad, f s s N. If the ca is dien fast enough,at a speed geate than the optimum speed, it may skid off up the incline so that f s is down the incline. Resole N and s f into two pependicula components : N cos etically up and f s sin etically down; N sin and f s cos hoizontally towads the cente of the cicula path. So long as the Cicula Motion Page 5

6 Shee Datta Coaching Classes, Contact No Pof. Deepak Jawale ca negotiates the tun without skidding off, the hoizontal components N sin and f s cos togethe poide the necessay centipetal foce, and N cos balances the sum mg + f s sin. If max is the maximum safe speed (without skidding), m max N sin + f s cos N sin + s N cos and N cos mg + f s sin N (sin + s cos ) () mg + s N sin mg N (cos s sin ) () Diiding () by (), m max / N (sin + s cos ) mg N (cos sin ) s max sin + s cos g cos sin tan + s tan s s max g(tan + s) tan s (3) This is the expession fo the maximum safe speed on a banked oad. At the optimum speed, the fiction between the ca tyes and the oad suface is no called into play. Hence, by setting s 0 in Eq. (3), the optimum speed on a banked cicula oad is. g tan Q.9 Define angle banking and state the factos on which it depends. Ans : Angle of Banking : The suface of the oad along the cue is kept inclined to the hoizontal at some suitable angle. So that the oute edge of the oad is at a highe eleation than the inne edge. This angle of inclination with the hoizontal is called as angle of banking. tan ( / g ) The factos on which angle of banking depends ae, ) Speed of ehicle. ) Radius of cuatue. 3) Acceleation due to gaity. Expession fo peiod of conical pendulum O An aangement consisting of a small heay sphee (bob), suspended by a light sting fom a igid suppot, is called a simple pendulum. Suppose that the pendulum is displaced in such a way that the bob moes with a constant speed along a hoizontal cicle. Duing such a motion, the pendulum descibes a cone in space. The simple pendulum used in this manne is called a conical pendulum. Let m be the mass of the bob and be the adius of the cicle along which the bob moes. Let l be the length of the pendulum. Duing the motion of the bob, the sting is inclined to Conical pendulum the etical at an angle. In any position of the bob, thee ae two foces acting upon it. These foces ae (i) its weight mg acting etically downwads and (ii) The tension T in the sting. The tension can be esoled into a etical component Tcos and a hoizontal component T sin. The component Tcos balances the weight of the bob. Tcos mg.. () The component Tsin poides the necessay centipetal foce fo UCM. Tsin m.. () Cicula Motion Page 6

7 Shee Datta Coaching Classes, Contact No Diiding eq. () by eq. (), we get tan g.. (3) Pof. Deepak Jawale g tan.. (4) This expession gies the magnitude of the elocity of the bob. tan AC.. (5) OC h Which h is the etical height of the conical pendulum. Fom eq. (3) and eq. (5), we get. g h h g h g The peiod of eolution (t) of the bob is gien by t t h g.. (6) Now, h l cos h lcos Theefoe Eq. (6) becomes lcos t.. (7) g This is the expession fo peiod of conical pendulum. Vetical cicula motion unde eath s gaitation Conside a body of mass m, is tied to one end of a sting and whiled in a etical cicle of adius. Duing its motion, the body is acted upon by two foces. i) its weight mg acting etically downwads and ii) the tension T in the sting, which is diected towads the cente of the cicle. Vtical Cicula Motion Suppose that the body is at point P, its weight mg can be esoled into two pependicula components :- (i) a component mg cos acting along OP and (ii) a component mg sin acting at ight angles to OP. The component mg sin opposes the upwad motion of the body, as a esult of which, the elocity of the body, goes on deceasing as it ises fom its lowest position. A to the highest position B. On the othe hand, duing the downwad motion of the body, the component mg sin acts in the same diection as that of the elocity of the body and hence the elocity of the body goes on inceasing. The aboe consideation shows that in the etical cicula motion, the elocity of the body goes on changing both in magnitude as well as in diection. In othe wods, the etical cicula motion is a nonunifom cicula motion. Equation fo elocity and enegy at diffeent positions of etical cicula motion Conside a body of mass m pefoming etical cicula motion, as we know that etical cicula motion is a nonunifom cicula motion in which the elocity of the body goes on changing continuously in both magnitude as well as diection. Cicula Motion Page 7

8 Shee Datta Coaching Classes, Contact No Pof. Deepak Jawale Let T a and T b be tensions in the sting at the points A and B espectiely. Let a and b be the coesponding elocities of the body at the points A and B. In any position of the body, the centipetal foce V T T acting on the body is gien by the geneal equation. d c d m T mg cos.. () At the lowest point A, T T a, a and 0 o, so that cos. Substituting these alues in eq. (), we get m a T a mg.. () At the highest point B, T T b, b and 80 o, so that cos Substituting these alues in eq. (), we get m b T b + mg.. (3) If the tension T b at the highest point is zeo, the entie centipetal foce is supplied by the weight of the body. Theefoe eq. (3) becomes. m b mg b g.. (4) This expession gie the minimum elocity which the body should possess at the highest point, so that the sting does not become slack. It the elocity at the highest point is less than g, the sting will become slack and the body will not be able to continue its cicula motion. The minimum elocity at the lowest point A can be found by applying the law of conseation of enegy. Let us assume that the potential enegy at the hoizontal leel passing though the point A is zeo. As the body moes fom the lowest point A to the highest point B, it is aised though a etical distance equal to. Hence, accoding to the law of conseation of enegy. K.E. at A K.E. at B + P.E. at B m a m b + mg () m a m b + a b + 4 g mg (4) Substituting the alue of b fom eq. (4), we get a g + 4g 5 g a 5g.. (5) This expession gies the minimum elocity which the body must possess at the lowest point, so that it can safely tael along the etical cicle of adius. Let us now find the elocity ( c ) of the body at the point C. In the position C, the body is aised aboe the point A though the etical distance. Hence the body possesses both K.E. and P.E. at the point C. Accoding to the law of conseation of enegy, K.E. at A K.E. at C + P.E. at C m a m c + mg m a m c + mg a c + g Cicula Motion Page 8

9 Shee Datta Coaching Classes, Contact No But a 5 g 5 g c + g c 3 g c 3 g.. (6) Pof. Deepak Jawale This expession gies the elocity of the body at the point C. The total enegy (E) of the body at the point A is E m a But a 5 g E 5 mg.. (7) The total enegy (E) of the body at the point B is E K.E. at B + P.E. at B m b + mg () E mg + mg 5 mg The total enegy (E) of the body at the point C is E K.E. at C + P.E. at C m c + mg E mg + mg 5 mg At any othe point of the etical cicle, the body possesses both K.E. and P.E.. Howee the total enegy at any point is same. Kinematical equations fo cicula motion in analogy with linea motion In cicula motion, we conside quantities such as angula displacement, angula elocity, angula acceleation, etc. These quantities ae analogous to the coesponding quantities in linea motion, such as linea displacement, linea elocity, linea acceleation, etc. Thus, angula displacement is analogous to linea displacement, angula elocity is analogous to linea elocity and angula acceleation is analogous to linea acceleation. If a paticle moing along a staight line with a linea elocity u is subjected to a unifom linea acceleation a, its linea elocity goes on changing with time. Let be its elocity afte time t and s be its linea displacement in time t. Then, u + at.. () s ut + at.. () and u + as (3) These equations ae called the kinematical equations fo unifomly acceleated nea motion. In analogy with linea motion, we can wite thee kinematical equations fo unifomly acceleated cicula motion, in the following manne. Conside a paticle moing along a cicle with an angula elocity 0. Suppose that the paticle is subjected to a unifom angula acceleation so that its angula elocity goes on changing with time. If is its angula elocity afte time t and is its angula displacement in time t, then the thee kinematical equations fo unifomly acceleated cicula motion can be stated as follows, 0 + t.. (4) 0 t + t.. (5) (6) Cicula Motion Page 9

10 Shee Datta Coaching Classes, Contact No Pof. Deepak Jawale Q0) In cicula motion, assuming ω, obtain an expession fo the esultant acceleation of a paticle in tems of tangential and adial component. Ans: Conside a paticle pefoming cicula motion, with linea elocity and angula elocity ω. We know that, ω. d dω + ω d dt dt dt d dt α + ω a a T + a. Fo Any Quey / Doubts Regading any online matte published by us students can diectly contact us. Shee Datta Coaching Classes, Fist Floo, Siddheshwa Ganpati Mandi, Z.P. Colony, Shasti Naga, Akola OR, Shee Datta Coaching Classes, Vadhaman Naga, Behind Vitthal-Rukmini Appt, Ring Road, Kaulkhed, Akola, Mob: , Mail us: ajanpatil@sdccakl.in isit us: Cicula Motion Page 0

r cos, and y r sin with the origin of coordinate system located at

r cos, and y r sin with the origin of coordinate system located at Lectue 3-3 Kinematics of Rotation Duing ou peious lectues we hae consideed diffeent examples of motion in one and seeal dimensions. But in each case the moing object was consideed as a paticle-like object,