Motion in Two Dimension (Circular Motion)

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1 Physics Motion in Two Dimension (Cicua Motion)

2 Tabe of Content 1. Vaiabes of cicua motion.. Centipeta acceeation. 3. Centipeta foce. 4. Centifuga foce. 5. Wok done by the centipeta foce. 6. Skidding of ehice on a ee oad. 7. Skidding of object on a otating patfom. 8. Bending of a cycist. 9. Banking of a oad. 10. Oetuning of ehice. 11. Motion of chaged patice in magnetic fied. 1. Reaction of oad on ca. 13. Non-unifom cicua motion. 14. Equations of cicua motion. 15. Motion in etica cice. 16. Conica penduum. 1

3 Cicua motion is anothe exampe of motion in two dimensions. To ceate cicua motion in a body it must be gien some initia eocity and a foce must then act on the body which is aways diected at ight anges to instantaneous eocity. Since this foce is aways at ight anges to the dispacement due to the initia eocity theefoe no wok is done by the foce on the patice. Hence, its kinetic enegy and thus speed is unaffected. But due to simutaneous action of the foce and the eocity the patice foows esutant path, which in this case is a cice. Cicua motion can be cassified into two types Unifom cicua motion and non-unifom cicua motion. F F F F CIRCULAR MOTION 1. Vaiabes of Cicua Motion. (1) Dispacement and distance: When patice moes in a cicua path descibing an ange duing time t (as shown in the figue) fom the position A to the position B, we see that the magnitude of the position ecto (that is equa to the adius of the cice) emains constant. i.e., diection of the position ecto changes fom time to time. (i) Dispacement: The change of position ecto o the dispacement A to the position B is gien by efeing the figue. 1 1 Putting 1 we obtain 1 1 cos 1 and the of the patice fom position O B A. cos B sin 1 cos sin O A

4 (ii) Distance: The distanced coeed by the patice duing the time t is gien as d = ength of the ac AB = (iii) Ratio of distance and dispacement: d sin / cosec ( / ) () Angua dispacement (): The ange tuned by a body moing on a cice fom some efeence ine is caed angua dispacement. (i) Dimension = [M 0 L 0 T 0 ] (as = ac / adius). (ii) Units = Radian o Degee. It is sometimes aso specified in tems of faction o mutipe of eoution. (iii) o ad 360 1Reoution (i) Angua dispacement is a axia ecto quantity. Its diection depends upon the sense of otation of the object and can be gien by Right Hand Rue; which states that if the cuatue of the finges of ight hand epesents the sense of otation of the object, then the thumb, hed pependicua to the cuatue of the finges, epesents the diection of angua dispacement ecto. () Reation between inea dispacement and angua dispacement o s s O S (3) Angua eocity (): Angua eocity of an object in cicua motion is defined as the time ate of change of its angua dispacement. (i) Angua eocity = d dt (ii) Dimension: [M 0 L 0 T 1 ] ange taced time taken Lt t 0 d t dt (iii) Units: Radians pe second (ad.s 1 ) o Degee pe second. (i) Angua eocity is an axia ecto. () Reation between angua eocity and inea eocity Its diection is the same as that of. Fo anticockwise otation of the point object on the cicua path, the diection of, accoding to Right hand ue is aong the axis of cicua path diected upwads. Fo cockwise otation of the point object on the cicua path, the diection of is aong the axis of cicua path diected downwads. 3

5 Note: It is impotant to note that nothing actuay moes in the diection of the angua eocity ecto. The diection of simpy epesents that the otationa motion is taking pace in a pane pependicua to it. (i) Fo unifom cicua motion emains constant wheeas fo non-unifom motion aies with espect to time. (4) Change in eocity: We want to know the magnitude and diection of the change in eocity of the patice which is pefoming unifom cicua motion as it moes fom A to B duing time t as shown in figue. The change in eocity ecto is gien as 1 o 1 cos Fo unifom cicua motion The diection of 1 So 1 cos 1 1 sin is shown in figue that can be gien as 180 o 90 o / Note: Reation between inea eocity and angua eocity. O B A In ecto fom (5) Time peiod (T): In cicua motion, the time peiod is defined as the time taken by the object to compete one eoution on its cicua path. (i) Units: second. (ii) Dimension: [M 0 L 0 T] (iii) Time peiod of second s hand of watch = 60 second. (i) Time peiod of minute s hand of watch = 60 minute () Time peiod of hou s hand of watch = 1 hou (6) Fequency (n): In cicua motion, the fequency is defined as the numbe of eoutions competed by the object on its cicua path in a unit time. (i) Units: s 1 o hetz (Hz). (ii) Dimension: [M 0 L 0 T 1 ] 4

6 Note: Reation between time peiod and fequency: If n is the fequency of eoution of an object in cicua motion, then the object competes n eoutions in 1 second. Theefoe, the object wi compete one eoution in 1/n second. T 1 / n Reation between angua eocity, fequency and time peiod: Conside a point object descibing a unifom cicua motion with fequency n and time peiod T. When the object competes one eoution, the ange taced at its axis of cicua motion is adians. It means, when time t = T, adians. Hence, angua eocity n ( T = 1/n) t T n T If two patices ae moing on same cice o diffeent copana concentic cices in same diection with diffeent unifom angua speeds A and B espectiey, the angua eocity of B eatie to A wi be e B A So the time taken by one to compete one eoution aound O with espect to the othe (i.e., time in which B compete one eoution aound O with espect to the othe (i.e., time in which B competes one moe o ess eoution aound O than A) T e 1 T1T T T 1 as T Specia case: If B A, e 0 and so T =., patices wi maintain thei position eatie to each othe. This is what actuay happens in case of geostationay sateite ( 1 = = constant) (7) Angua acceeation (): Angua acceeation of an object in cicua motion is defined as the time ate of change of its angua eocity. (i) If be the change in angua eocity of the object in time intea t and t + t, whie moing on a cicua path, then angua acceeation of the object wi be Lt (ii) Units: ad. s (iii) Dimension: [M 0 L 0 T ] d t dt t0 d dt (i) Reation between inea acceeation and angua acceeation a 5

7 d () Fo unifom cicua motion since is constant so 0 dt (i) Fo non-unifom cicua motion 0 Note: Reation between inea (tangentia) acceeation and angua acceeation a a. Fo unifom cicua motion angua acceeation is zeo, so tangentia acceeation aso is equa to zeo. b. Fo non-unifom cicua motion a 0 (because 0).. Centipeta Acceeation. (1) Acceeation acting on the object undegoing unifom cicua motion is caed centipeta acceeation. () It aways acts on the object aong the adius towads the cente of the cicua path. 4 (3) Magnitude of centipeta acceeation a 4n T (4) Diection of centipeta acceeation: It is aways the same as that of. When t deceases, aso deceases. Due to which becomes moe and moe pependicua to. When t 0, becomes pependicua to the eocity ecto. As the eocity ecto of the patice at an instant acts aong the tangent to the cicua path, theefoe and hence the centipeta acceeation ecto acts aong the adius of the cicua path at that point and is diected towads the cente of the cicua path. ac 6

8 3. Centipeta Foce. Accoding to Newton's fist aw of motion, whenee a body moes in a staight ine with unifom eocity, no foce is equied to maintain this eocity. But when a body moes aong a cicua path with unifom speed, its diection changes continuousy i.e. eocity keeps on changing on account of a change in diection. Accoding to Newton's second aw of motion, a change in the diection of motion of the body can take pace ony if some extena foce acts on the body. Due to inetia, at eey point of the cicua path; the body tends to moe aong the tangent to the cicua path at that point (in figue). Since eeybody has diectiona inetia, a eocity cannot change by itsef and as such we hae to appy a foce. But this foce shoud be such that it changes the diection of eocity and not its magnitude. This is possibe ony if the foce acts pependicua to the diection of eocity. Because the eocity is aong the tangent, this foce must be aong the adius (because the adius of a cice at any point is pependicua to the tangent at that point). Futhe, as this foce is to moe the body in a cicua path, it must acts towads the cente. This cente-seeking foce is caed the centipeta foce. Hence, centipeta foce is that foce which is equied to moe a body in a cicua path with unifom speed. The foce acts on the body aong the adius and towads cente. (1) Fomuae fo centipeta foce: F () Centipeta foce in diffeent situation m m 4 n m4 T F F F F Situation A patice tied to a sting and whied in a hoizonta cice Vehice taking a tun on a ee oad A ehice on a speed beake Reoution of eath aound the sun Centipeta Foce Tension in the sting Fictiona foce exeted by the oad on the tyes Weight of the body o a component of weight Gaitationa foce exeted by the sun 7

9 Eecton eoing aound the nuceus in an atom A chaged patice descibing a cicua path in a magnetic fied Couomb attaction exeted by the potons in the nuceus Magnetic foce exeted by the agent that sets up the magnetic fied 4. Centifuga Foce. It is an imaginay foce due to incopoated effects of inetia. When a body is otating in a cicua path and the centipeta foce anishes, the body woud eae the cicua path. To an obsee A who is not shaing the motion aong the cicua path, the body appeas to fy off tangentia at the point of eease. To anothe obsee B, who is shaing the motion aong the cicua path (i.e., the obsee B is aso otating with the body with the same eocity), the body appeas to be stationay befoe it is eeased. When the body is eeased, it appeas to B, as if it has been thown off aong the adius away fom the cente by some foce. In eaity no foce is actuay seen to act on the body. In absence of any ea foce the body tends to continue its motion in a staight ine due to its inetia. The obsee A easiy eates this eents to be due to inetia but since the inetia of both the obsee B and the body is same, the obsee B cannot eate the aboe happening to inetia. When the centipeta foce ceases to act on the body, the body eaes its cicua path and continues to moes in its staight-ine motion but to obsee B it appeas that a ea foce has actuay acted on the body and is esponsibe fo thowing the body adiay out-wods. This imaginay foce is gien a name to expain the effects on inetia to the obsee who is shaing the cicua motion of the body. This inetia foce is caed centifuga foce. Thus centifuga foce is a fictitious foce which has significance ony in a otating fame of efeence. 8

10 5. Wok done by Centipeta Foce. The wok done by centipeta foce is aways zeo as it is pependicua to eocity and hence instantaneous dispacement. Wok done = Incement in kinetic enegy of eoing body S Wok done = 0 Aso W = F. S = F S cos 90 o F = FS cos 90 o = 0 Exampe: (i) When an eecton eoe aound the nuceus in hydogen atom in a paticua obit, it neithe absob no emit any enegy means its enegy emains constant. (ii) When a sateite estabished once in a obit aound the eath and it stats eoing with paticua speed, then no fue is equied fo its cicua motion. 6. Skidding of Vehice on a Lee Road. When a ehice tuns on a cicua path it equies centipeta foce. If fiction poides this centipeta foce then ehice can moe in cicua path safey if Fiction foce equied centipeta foce mg mg m safe g This is the maximum speed by which ehice can tun in a cicua path of adius, whee coefficient of fiction between the oad and tye is. 9

11 7. Skidding of Object on a Rotating Patfom. On a otating patfom, to aoid the skidding of an object (mass m) paced at a distance fom axis of otation, the centipeta foce shoud be poided by foce of fiction. Centipeta foce = Foce of fiction m = mg ( g / ), max Hence maximum angua eocity of otation of the patfom is ( g / ), so that object wi not skid on it. 8. Bending of a Cycist. A cycist poides himsef the necessay centipeta foce by eaning inwad on a hoizonta tack, whie going ound a cue. Conside a cycist of weight mg taking a tun of adius with eocity. In ode to poide the necessay centipeta foce, the cycist eans though ange inwads as shown in figue. The cycist is unde the action of the foowing foces: The weight mg acting eticay downwad at the cente of gaity of cyce and the cycist. The eaction R of the gound on cycist. It wi act aong a ine-making ange with the etica. The etica component R cos of the noma eaction R wi baance the weight of the cycist, whie the hoizonta component R sin wi poide the necessay centipeta foce to the cycist. R sin..(i) and R cos = mg..(ii) R R cos Diiding equation (i) by (ii), we hae / R sin o R sin m R cos mg tan.. (iii) g mg 10

12 Theefoe, the cycist shoud bend though an ange tan 1 g It foows that the ange though which cycist shoud bend wi be geate, if (i) The adius of the cue is sma i.e. the cue is shape (ii) The eocity of the cycist is age. Note: Fo the same easons, an ice skate o an aipane has to bend inwads, whie taking a tun. 9. Banking of a Road. Fo getting a centipeta foce cycist bend towads the cente of cicua path but it is not possibe in case of fou wheees. Theefoe, oute bed of the oad is aised so that a ehice moing on it gets automaticay incined towads the cente. In the figue (A) shown eaction R is esoed into two components, the component R cos baances weight of ehice R cos mg (i) and the hoizonta component R sin poides necessay centipeta foce as it is diected towads cente of desied cice Thus R sin... (ii) Diiding (ii) by (i), we hae o tan... (iii) g tan... (i) [As = ] g g R cos R R sin x mg Fig. (A) Fig. (B) h If = width of the oad, h = height of the oute edge fom the gound ee then fom the figue (B) 11

13 h h tan...() [since is ey sma] x Fom equation (iii), (i) and () tan g g g h Note: a. If fiction is aso pesent between the tyes and oad then b. Maximum safe speed on a banked fictiona oad tan g 1 tan g ( tan ) 1 tan 10. Oetuning of Vehice. When a ca moes in a cicua path with speed moe than maximum speed then it oetuns and it s inne whee eaes the gound fist Weight of the ca = mg Speed of the ca = Radius of the cicua path = Distance between the cente of whees of the ca = a Height of the cente of gaity (G) of the ca fom the oad ee = h Reaction on the inne whee of the ca by the gound = R 1 Reaction on the oute whee of the ca by the gound = R When a ca moe in a cicua path, hoizonta foce F poides the equied centipeta foce i.e., F...(i) R Fo otationa equiibium, by taking the moment of foces R 1, R and F about G Fh R1a Ra...(ii) As thee is no etica motion so R 1 + R = mg By soing (i), (ii) and (iii) R 1 1 h M g a...(iii)...(i) F R1 h G a mg R 1

14 and R 1 h M g a...() It is cea fom equation (i) that if inceases aue of R 1 deceases and fo R 1 = 0 h g a o ga h i.e. the maximum speed of a ca without oetuning on a fat oad is gien by ga h 11. Motion of Chaged Patice in Magnetic Fied. When a chaged patice haing mass m, chage q entes pependicuay in a magnetic fied B, with eocity then it descibes a cicua path of adius. Because magnetic foce (qb) woks in the pependicua diection of and it poides equied centipeta foce Magnetic foce = Centipeta foce qb = adius of the cicua path qb F q 13

15 1. Reaction of Road on Ca. (1) When ca moes on a concae bidge then Centipeta foce = and eaction R mgcos R mgcos mg cos mg R Concae bidge () When ca moes on a conex bidge Centipeta foce = mgcos R and eaction R mgcos R mg cos mg Conex bidge 13. Non-Unifom Cicua Motion. If the speed of the patice in a hoizonta cicua motion changes with espect to time, then its motion is said to be non-unifom cicua motion. Conside a patice descibing a cicua path of adius with cente at O. Let at an instant the patice be at P and be its inea eocity and be its angua eocity. Then,..(i) Diffeentiating both sides of w..t. time t we hae d d d dt dt dt d..(ii) Hee, a, (Resutant acceeation) dt d a (Angua acceeation) dt d a a t a c..(iii) dt (Linea eocity) Thus the esutant acceeation of the patice at P has two component acceeations a O ac at P 14

16 (1) Tangentia acceeation: a t It acts aong the tangent to the cicua path at P in the pane of cicua path. Accoding to ight hand ue since and ae pependicua to each othe, theefoe, the magnitude of tangentia acceeation is gien by at sin 90. () Centipeta (Radia) acceeation: o a c It is aso caed centipeta acceeation of the patice at P. It acts aong the adius of the patice at P. Accoding to ight hand ue since and ae pependicua to each othe, theefoe, the magnitude of centipeta acceeation is gien by o a sin 90 = ( ) / c (3) Tangentia and centipeta acceeation in diffeent motions Centipeta acceeation Tangentia acceeation Net acceeation Type of motion a c = 0 a t = 0 a = 0 Unifom tansatoy motion a c = 0 a t 0 a = a t Acceeated tansatoy motion a c 0 a t = 0 a = a c Unifom cicua motion a c 0 a t 0 a a c a t Non-unifom cicua motion Note: Hee a t goens the magnitude of whie a c its diection of motion. (4) Foce: In non-unifom cicua motion the patice simutaneousy possesses two foces Centipeta foce: F ma Tangentia foce: F ma t c t c m 15

17 Net foce: Fnet ma = m a c a t Note: In non-unifom cicua motion wok done by centipeta foce wi be zeo since F c a. In non-unifom cicua motion wok done by tangentia of foce wi not be zeo since F t 0 b. Rate of wok done by net foce in non-unifom cicua = ate of wok done by tangentia foce i.e. P dw dt F. t 14. Equations of Cicua Motion. Fo acceeated motion Fo etaded motion 1 t 1 t 1 1t t 1 1t t 1 1 Whee 1 = Initia angua eocity of patice = Fina angua eocity of patice = Angua acceeation of patice = Ange coeed by the patice in time t n = Ange coeed by the patice in n th second n 1 (n 1) n 1 (n 1) 15. Motion in Vetica Cice. This is an exampe of non-unifom cicua motion. In this motion body is unde the infuence of gaity of eath. When body moes fom owest point to highest point. Its speed decease and becomes minimum at highest point. Tota mechanica enegy of the body emains conseed and KE conets into PE and ice esa. (1) Veocity at any point on etica oop: If u is the initia eocity impated to body at owest point then. Veocity of body at height h is gien by C 16 D O h A u B P

u gh u g(1 cos) [As h = cos = (1 cos)] Whee in the ength of the sting () Tension at any point on etica oop: Tension at genea point P, Accoding to Newton s second aw of motion.

18 u gh u g(1 cos) [As h = cos = (1 cos)] Whee in the ength of the sting () Tension at any point on etica oop: Tension at genea point P, Accoding to Newton s second aw of motion. Net foce towads cente = centipeta foce T mgcos O T mgcos m [ T u g( 3 cos)] [As u g(1 cos) ] D O C A T mg B P mg cos + / (3) Veocity and tension in a etica oop at diffeent positions Position Ange Veocity Tension A 0 o mu u mg B 90 o u mu g mg C 180 o u mu 4g 5mg D 70 o u mu g mg It is cea fom the tabe that: and T T A A T T and T B = T D T B B C 3mg, TA TC 6mg TB TC 3mg 17

19 (4) Vaious conditions fo etica motion: Veocity at owest point Condition u A 5g Tension in the sting wi not be zeo at any of the point and body wi continue the cicua motion. u A 5g, Tension at highest point C wi be zeo and body wi just compete the cice. g u 5g, Patice wi not foow cicua motion. Tension in sting become A zeo somewhee between points B and C wheeas eocity emain positie. Patice eaes cicua path and foow paaboic tajectoy. u A g Both eocity and tension in the sting becomes zeo between A and B and patice wi osciate aong semi-cicua path. u A g eocity of patice becomes zeo between A and B but tension wi not be zeo and the patice wi osciate about the point A. Note: K.E. of a body moing in hoizonta cice is same thoughout the path but the K.E. of the body moing in etica cice is diffeent at diffeent paces. If body of mass m is tied to a sting of ength and is pojected with a hoizonta eocity u then: u Height at which the eocity anishes is h g Height at which the tension anishes is u g h 3g (5) Citica condition fo etica ooping: If the tension at C is zeo, then body wi just compete eoution in the etica cice. This state of body is known as citica state. The speed of body in citica state is caed as citica speed. mu Fom the aboe tabe T C = 5mg 0 u 5g It means to compete the etica cice the body must be pojected with minimum eocity of at the owest point. 5 g 18

20 (6) Vaious quantities fo a citica condition in a etica oop at diffeent positions : Quantity Point A Point B Point C Point D Point P Linea eocity () 5 g 3 g g 3 g g ( 3 cos ) Angua eocity () 5g 3g g 3g g ( 3 cos ) Tension in Sting (T) Kinetic Enegy (KE) Potentia Enegy (PE) Tota Enegy (TE) 6 mg 3 mg 0 3 mg 3mg (1 cos ) 5 mg 3 mg 1 mg 3 mg mg (3 cos ) 0 mg mg mg mg ( 1 cos ) 5 mg 5 mg 5 mg 5 mg 5 mg (7) Motion of a bock on fictioness hemisphee: A sma bock of mass m sides down fom the top of a fictioness hemisphee of adius. The component of the foce of gaity (mg cos) poides equied centipeta foce but at point B it's cicua motion ceases and the bock ose contact with the suface of the sphee. Fo point B, by equating the foces, mgcos...(i) Fo point A and B, by aw of conseation of enegy Tota enegy at point A = Tota enegy at point B A B ( h) h K.E. (A) + P.E. (A) = K.E. (B) + P.E. (B) mg = mgh and fom the gien figue g( h)...(ii) h cos...(iii) By substituting the aue of and h fom eq n (ii) and (iii) in eq n (i) h mg m g( h) h ( h) h 3 i.e. the bock ose contact at the height of and ange fom the etica can be gien by 3 fom the gound. h cos cos mg 19

21 16. Conica Penduum. This is the exampe of unifom cicua motion in hoizonta pane. A bob of mass m attached to a ight and in-extensibe sting otates in a hoizonta cice of adius with constant angua speed about the etica. The sting makes ange with etica and appeas tacing the suface of a cone. So this aangement is caed conica penduum. The foce acting on the bob ae tension and weight of the bob. S Fom the figue and (1) Tension in the sting: mg T cos T sin.(i) T cos mg.(ii) T mg mg () Ange of sting fom the etica: 1 g [As tan (3) Linea eocity of the bob: gtan h cos ] g T T sin O S h O T cos P / mg P (4) Angua eocity of the bob: g tan g h g cos (5) Time peiod of eoution: T P cos g h g g g tan 0

DYNAMICS OF UNIFORM CIRCULAR MOTION

DYNAMICS OF UNIFORM CIRCULAR MOTION Chapte 5 Dynamics of Unifom Cicula Motion Chapte 5 DYNAMICS OF UNIFOM CICULA MOTION PEVIEW An object which is moing in a cicula path with a constant speed is said to be in unifom cicula motion. Fo an object