2. Kinematics. 2.1 Motion in One Dimension: Position

Transcription

1 1 Motion in ne Dimension: osition. Kinematics osition of any point is completely expessed by two factos: Its distance fom the obsee and its diection with espect to obsee. hat is why position is chaacteised by a ecto known as position ecto. Let point is in a xy plane and its coodinates ae (x, y). hen position ecto ( ) of point will be xiˆ + yj ˆ and if the point is in a space and its coodinates ae (x, y, z) then position ecto can be expessed as = xiˆ + yj ˆ + zkˆ.. Rest and Motion. If a body does not chane its position as time passes with espect to fame of efeence, it is said to be at est. And if a body chanes its position as time passes with espect to fame of efeence, it is said to be in motion. Fame of Refeence: It is a system to which a set of coodinates ae attached and with efeence to which obsee descibes any eent. Rest and motion ae elatie tems. It depends upon the fame of efeences..3 ypes of Motion. ne dimensional wo dimensional hee dimensional Motion of a body in a staiht line is called one dimensional motion. When only one coodinate of the position of a body chanes with time then it is said to be moin one dimensionally. e... Motion of ca on a staiht oad. Motion of feely fallin body..4 aticle o oint Mass. Motion of body in a plane is called two dimensional motion. When two coodinates of the position of a body chanes with time then it is said to be moin two dimensionally. e.. Motion of ca on a cicula tun. Motion of billiads ball. Motion of body in a space is called thee dimensional motion. When all thee coodinates of the position of a body chanes with time then it is said to be moin thee dimensionally. e... Motion of flyin kite. Motion of flyin insect. he smallest pat of matte with zeo dimension which can be descibed by its mass and position is defined as a paticle. If the size of a body is neliible in compaison to its ane of motion then that body is known as a paticle..5 Distance and Displacement (1) Distance: It is the actual path lenth coeed by a moin paticle in a ien inteal of time. (i) Its a scala quantity. (ii) Dimension: [M 0 L 1 0 ] (iii) Unit: mete (S.I.)

2 Kinematics / () Displacement: Displacement is the chane in position ecto i.e., A ecto joinin initial to final position. (i) Displacement is a ecto quantity (ii) Dimension: [M 0 L 1 0 ] (iii) Unit: mete (S.I.) (i) If S1, S, S3... Sn ae the displacements of a body then the total (net) displacement is the ecto sum of the indiiduals. S = S1 + S + S Sn (3) Compaison between distance and displacement: (i) Distance Displacement. (ii) Fo a moin paticle distance can nee be neatie o zeo while displacement can be. i.e., Distance > 0 but Displacement > = o < 0 (iii) Fo motion between two points displacement is sinle alued while distance depends on actual path and so can hae many alues. (i) Fo a moin paticle distance can nee decease with time while displacement can. Decease in displacement with time means body is moin towads the initial position. () In eneal manitude of displacement is not equal to distance. Howee, it can be so if the motion is alon a staiht line without chane in diection..6 Speed and Velocity. (1) Speed: Rate of distance coeed with time is called speed. (i) It is a scala quantity hain symbol υ. (ii) Dimension: [M 0 L 1 1 ] (iii) Unit: mete/second (S.I.), cm/second (C.G.S.) (i) ypes of speed: (a) Unifom speed: When a paticle coes equal distances in equal inteals of time, (no matte how small the inteals ae) then it is said to be moin with unifom speed. (b) Non-unifom (aiable) speed: In non-unifom speed paticle coes unequal distances in equal inteals of time. (c) Aeae speed: he aeae speed of a paticle fo a ien Inteal of time is defined as the atio of distance taelled to the time taken. Distance taelled s Aeae speed = ; a = ime taken t ime aeae speed: When paticle moes with diffeent unifom speed υ1, υ, υ3... etc in diffeent time inteals t1, t, t3,... etc espectiely, its aeae speed oe the total time of jouney is ien as otal distance coeed d1 + d + d υ 1t1 + υ t + υ 3t3 +. a = = = otal time elapsed t + t + t +... t + t + t Special case: When paticle moes with speed 1 upto half time of its total 1 + motion and in est time it is moin with speed then a = Distance aeaed speed: When a paticle descibes diffeent distances d1, d, d3,.... with diffeent time inteals t1, t, t3,... with speeds 1,, 3,..

3 3 espectiely then the speed of paticle aeaed oe the total distance can be ien as otal distance coeed d1 + d + d d1 + d + d otal time elapsed t1 + t + t d1 d d υ υ υ 1 3 (d) Instantaneous speed: It is the speed of a paticle at paticula instant. When we say speed, it usually means instantaneous speed. he instantaneous speed is aeae speed fo infinitesimally small time inteal (i.e., t 0). hus s ds Instantaneous speed = lim = t 0 t dt () Velocity: Rate of chane of position i.e. ate of displacement with time is called elocity. (i) It is a scala quantity hain symbol. (ii) Dimension: [M 0 L 1 1 ] (iii) Unit: mete/second (S.I.), cm/second (C.G.S.) (i) ypes (a) Unifom elocity: A paticle is said to hae unifom elocity, if manitudes as well as diection of its elocity emains same and this is possible only when the paticles moes in same staiht line without eesin its diection. (b) Non-unifom elocity: A paticle is said to hae non-unifom elocity, if eithe of manitude o diection of elocity chanes (o both chanes). (c) Aeae elocity: It is defined as the atio of displacement to time taken by the body Aeae elocity = Displacement ime taken ; = a t (d) Instantaneous elocity: Instantaneous elocity is defined as ate of chane of position ecto of paticles with time at a cetain instant of time. d Instantaneous elocity = lim = t 0 t dt () Compaison between instantaneous speed and instantaneous elocity (a) instantaneous elocity is always tanential to the path followed by the paticle. (b) A paticle may hae constant instantaneous speed but aiable instantaneous elocity. (c) he manitude of instantaneous elocity is equal to the instantaneous speed. (d) If a paticle is moin with constant elocity then its aeae elocity and instantaneous elocity ae always equal. (e) If displacement is ien as a function of time, then time deiatie of displacement will ie elocity. (i) Compaison between aeae speed and aeae elocity (a) Aeae speed is scala while aeae elocity is a ecto both hain same units (m/s) and dimensions [L 1 ]. (b) Aeae speed o elocity depends on time inteal oe which it is defined. (c) Fo a ien time inteal aeae elocity is sinle alued while aeae speed can hae many alues dependin on path followed.

4 a Kinematics / 4 (d) If afte motion body comes back to its initial position then = 0 (as = 0 ) but a > 0 and finite as ( s > 0). (e) Fo a moin body aeae speed can nee be neatie o zeo (unless t ) while aeae elocity can be i.e. a > 0 while a υ = o < 0..7 Acceleation. he time ate of chane of elocity of an object is called acceleation of the object. (1) It is a ecto quantity. It s diection is same as that of chane in elocity (Not of the elocity) () hee ae thee possible ways by which chane in elocity may occu When only diection of elocity chanes Acceleation pependicula to elocity e.. Unifom cicula motion When only manitude of elocity chanes Acceleation paallel o anti-paallel to elocity e.. Motion unde aity When both manitude and diection of elocity chanes Acceleation has two components one is pependicula to elocity and anothe paallel o antipaallel to elocity e.. ojectile motion (3) Dimension: [M 0 L 1 ] (4) Unit: mete/second (S.I.); cm/second (C.G.S.) (5) ypes of acceleation: (i) Unifom acceleation: A body is said to hae unifom acceleation if manitude and diection of the acceleation emains constant duin paticle motion. If a paticle is moin with unifom acceleation, this does not necessaily imply that paticle is moin in staiht line. e.. ojectile motion. (ii) Non-unifom acceleation: A body is said to hae non-unifom acceleation, if manitude o diection o both, chane duin motion. 1 (iii) Aeae acceleation: aa υ = = t t he diection of aeae acceleation ecto is the diection of the chane in elocity ecto as a = t d (i) Instantaneous acceleation = a = lim = t 0 t dt () Fo a moin body thee is no elation between the diection of instantaneous elocity and diection of acceleation. e.. (a) In unifom cicula motion = 90º always (b) In a pojectile motion is aiable fo eey point of tajectoy.

5 5 d d x dx (i) By definition a = = As = dt dt dt (ii) If elocity is ien as a function of position, then by chain ule d d dx dυ dx a = = = as dt dx dt dx = dt (iii) If a paticle is acceleated fo a time t1 by acceleation a1 and fo time t by a1t1 + at acceleation a then aeae acceleation is aa υ = t + t (ix) Acceleation can be positie, zeo o neatie. ositie acceleation means elocity inceasin with time, zeo acceleation means elocity is unifom constant while neatie acceleation (etadation) means elocity is deceasin with time. (x) Fo motion of a body unde aity, acceleation will be equal to, whee is the acceleation due to aity. Its nomal alue is 9.8 m/s o 980 cm/s o 3 feet/s..8 osition ime Gaph Vaious position time aphs and thei intepetation 1 = 0 o so = 0 i.e., line paallel to time axis epesents that the paticle is at est. = 90 o so = i.e., line pependicula to time axis epesents that paticle is chanin its position but time does not chanes it means the paticle possesses infinite elocity. actically this is not possible. = constant so = constant, a = 0 i.e., line with constant slope epesents unifom elocity of the paticle. is inceasin so is inceasin, a is positie. i.e., line bendin towads position axis epesents inceasin elocity of paticle. It means the paticle possesses acceleation. is deceasin so is deceasin, a is neatie i.e., line bendin towads time axis epesents deceasin elocity of the paticle. It means the paticle possesses etadation.

6 Kinematics / 6 constant but > 90 o so will be constant but neatie i.e., line with neatie slope epesent that paticle etuns towads the point of efeence. (neatie displacement). A B C Staiht line sements of diffeent slopes epesent that elocity of the body chanes afte cetain inteal of time. S his aph shows that at one instant the paticle has two positions. Which is not possible. he aph shows that paticle comin towads oiin initially and afte that it is moin away fom oiin. Note: If the aph is plotted between distance and time then it is always an inceasin cue and it nee comes back towads oiin because distance nee decease with time. Fo two paticles hain displacement time aph with slopes 1 and possesses elocities 1 and espectiely then υ = 1 tan 1 υ tan.9 Velocity ime Gaph. he aph is plotted by takin time t alon x-axis and elocity of the paticle on y-axis. Distance and displacement: he aea coeed between the elocity time aph and time axis ies the displacement and distance taelled by the body fo a ien time inteal. hen otal distance = Addition of modulus of diffeent aea. i.e. s = υ dt otal displacement = Addition of diffeent aea considein thei sin. i.e. = υdt Acceleation: It is clea that slope of elocity-time aph epesents the acceleation of the paticle.

7 7 Vaious elocity time aphs and thei intepetation Velocity = 0, a = 0, = constant ime i.e., line paallel to time axis epesents that the paticle is moin with constant elocity. Velocity ime = 90 o, a =, = inceasin i.e., line pependicula to time axis epesents that the paticle is inceasin its elocity, but time does not chane. It means the paticle possesses infinite acceleation. actically it is not possible. Velocity ime =constant, so a = constant and is inceasin unifomly with time i.e., line with constant slope epesents unifom acceleation of the paticle. Velocity inceasin so acceleation inceasin i.e., line bendin towads elocity axis epesent the inceasin ime acceleation in the body. Velocity ime deceasin so acceleation deceasin i.e. line bendin towads time axis epesents the deceasin acceleation in the body Velocity ime ositie constant acceleation because is constant and < 90 o but initial elocity of the paticle is neatie. Velocity ime ositie constant acceleation because is constant and < 90 o but initial elocity of paticle is positie.

8 Kinematics / 8 Velocity ime Neatie constant acceleation because is constant and > 90 o but initial elocity of the paticle is positie. Velocity ime Neatie constant acceleation because is constant and > 90 o but initial elocity of the paticle is zeo. Velocity ime Neatie constant acceleation because is constant and > 90 o but initial elocity of the paticle is neatie..10 Equations of Kinematics. hese ae the aious elations between u,, a, t and s fo the moin paticle whee the notations ae used as: u = Initial elocity of the paticle at time t = 0 sec = Final elocity at time t sec a = Acceleation of the paticle s = Distance taelled in time t sec sn = Distance taelled by the body in n th sec (1) When paticle moes with constant acceleation (i) Acceleation is said to be constant when both the manitude and diection of acceleation emain constant. (ii) hee will be one dimensional motion if initial elocity and acceleation ae paallel o anti-paallel to each othe. (iii) Equations of motion in scala fom Equation of motion in ecto fom υ = u + at = u + at 1 1 s = ut + at s = ut + at υ = u + as. u. u = a. s u + 1 s = t s = ( u + ) t a a sn = u + (n 1) sn = u + (n 1) () Impotant points fo unifomly acceleated motion (i) If a body stats fom est and moes with unifom acceleation then distance coeed by the body in t sec is popotional to t (i.e. s t ). So the atio of distance coeed in 1 sec, sec and 3 sec is 1 : :3 o 1: 4: 9. (ii) If a body stats fom est and moes with unifom acceleation then distance coeed by the body in nth sec is popotional to (n 1) (i.e. sn (n 1). So the atio of distance coeed in I sec, II sec and III sec is 1: 3: 5.

9 9 (iii) A body moin with a elocity u is stopped by application of bakes afte coein a distance s. If the same body moes with elocity nu and same bakin foce is applied on it then it will come to est afte coein a distance of n s..11 Motion of Body Unde Gaity (Fee Fall). Acceleation poduced in the body by the foce of aity, is called acceleation due to aity. It is epesented by the symbol. In the absence of ai esistance, it is found that all bodies fall with the same acceleation nea the suface of the eath. his motion of a body fallin towads the eath fom a small altitude (h << R) is called fee fall. An ideal one-dimensional motion unde aity in which ai esistance and the small chanes in acceleation with heiht ae nelected. (1) If a body dopped fom some heiht (initial elocity zeo) (i) Equation of motion: akin initial position as oiin and diection of motion (i.e., downwad diection) as a positie, hee we hae u = 0 [As body stats fom est] a = + [As acceleation is in the diection of motion] = t (i) 1 h = t (ii) υ = h (iii) hn = (n 1)...(i) (ii) Gaph of distance elocity and acceleation with espect to time: s a tan = t t t () If a body is pojected etically downwad with some initial elocity Equation of motion: υ = u + t 1 h = ut + t υ = u + h hn = u + (n 1) (3) If a body is pojected etically upwad (i) if the body is pojected with elocity u and afte time t it eaches up to heiht h then 1 υ = u t; h = ut t ; υ z = u h; hn = u (n 1)

10 Kinematics / 10 (ii) Fo maximum heiht = 0 So fom aboe equation u = t, 1 h = t and u = h (iii) Gaph of distance, elocity and acceleation with espect to time (fo maximum heiht): s (u /) a + (u/) (u/) t + t (u/) t a It is clea that both quantities do not depend upon the mass of the body o we can say that in absence of ai esistance, all bodies fall on the suface of the eath with the same ate. (4) he motion is independent of the mass of the body, as in any equation of motion, mass is not inoled. hat is why a heay and liht body when eleased fom the same heiht, each the ound simultaneously and with same elocity i.e., t = ( h / ) and = h. (6) In case of motion unde aity time taken to o up is equal to the time taken to fall down thouh the same distance. (7) In case of motion unde aity, the speed with which a body is pojected up is equal to the speed with which it comes back to the point of pojection. (8) A ball is dopped fom a buildin of heiht h and it eaches afte t seconds on eath. Fom the same buildin if two ball ae thown (one upwads and othe downwads) with the same elocity u and they each the eath suface afte t1 and t seconds espectiely then t = t t 1 (9) A body is thown etically upwads. If ai esistance is to be taken into account, then the time of ascent is less than the time of descent. t > t1.1 Motion with Vaiable Acceleation (i) If acceleation is a function of time a = f(t), then u f ( t) dt t = + and ( ( ) ) 0 (ii) If acceleation is a function of distance x a = f(x) then = u + f ( x) dx x0 s ut f t dt dt = + (iii) If acceleation is a function of elocity d a = f () then t = and x x0 u f ( ) = + u d f ( )

11 11 he motion of an object is called two dimensional, if two of the thee co-odinates ae equied to specify the position of the object in space chanes w..t time. In such a motion, the object moes in a plane. Fo example, a billiad ball moin oe the billiad table, an insect cawlin oe the floo of a oom, eath eolin aound the sun etc. wo special cases of motion in two dimension ae: 1. ojectile motion. Cicula motion RJECILE MIN.13 Intoduction. If the foce actin on a paticle is oblique with initial elocity then the motion of paticle is called pojectile motion..14 ojectile A body which is in fliht thouh the atmosphee but is not bein popelled by any fuel is called pojectile..15 Assumptions of ojectile Motion. (1) hee is no esistance due to ai. () he effect due to cuatue of eath is neliible. (3) he effect due to otation of eath is neliible. (4) Fo all points of the tajectoy, the acceleation due to aity is constant in manitude and diection..16 inciples of hysical Independence of Motions. (1) he motion of a pojectile is a two-dimensional motion. So, it can be discussed in two pats. Hoizontal motion and etical motion. hese two motions take place independent of each othe. his is called the pinciple of physical independence of motions. () he elocity of the paticle can be esoled into two mutually pependicula components. Hoizontal component and etical component. (3) he hoizontal component emains unchaned thouhout the fliht. he foce of aity continuously affects the etical component. (4) he hoizontal motion is a unifom motion and the etical motion is a unifomly acceleated etaded motion..17 ypes of ojectile Motion (1) blique pojectile motion () Hoizontal pojectile motion (3) ojectile motion on an inclined plane.18 blique ojectile. In pojectile motion, hoizontal component of elocity (u cos), acceleation () and mechanical eney emains constant while, speed, elocity, etical component of elocity (u sin ), momentum, kinetic eney and potential eney all chanes. Velocity, and KE ae maximum at the point of pojection while minimum (but not zeo) at hihest point. (1) Equation of tajectoy: A pojectile thown with elocity u at an anle with the hoizontal. Y x u y u sin u cos X

12 Kinematics / 1 Note: he elocity u can be esoled into two ectanula components u cos component alon 1 x X axis and u sin component alon Y axis. y = xtan u cos Equation of oblique pojectile also can be witten as x y = xtan 1 R (whee R = hoizontal ane ) () Displacement of pojectile ( ) : Let the paticle acquies a position hain the coodinates (x, y) just afte time t fom the instant of pojection. he coespondin position ecto of the paticle at time t is as shown in the fiue. = xiˆ + yj ˆ.(i) he hoizontal distance coeed duin time t is ien as x = t x = ucos t.(ii) x he etical elocity of the paticle at time t is ien as y = usin t 1/ t.(iii) and φ = 1 tan ( y / x) Note: he anle of eleation φ of the hihest point of the pojectile and the anle of 1 pojection ae elated to each othe as tan φ = tan (3) Instantaneous elocity : In pojectile motion, etical component of elocity chanes but hoizontal component of elocity emains always constant. Let i be the instantaneous elocity of pojectile at time t diection of this elocity is alon the tanent to the tajectoy at point. = i + ˆj = + i x y i x y y usin t Diection of instantaneous elocity tan α = = ucos (7) ime of fliht: he total time taken by the pojectile to o up and come down to the same leel fom which it was pojected is called time of fliht. Fo etical upwad motion 0 = u sin t t = (u sin /) usin ime of fliht = t = (8) Hoizontal ane: It is the hoizontal distance taelled by a body duin the time of fliht. So by usin second equation of motion R = u cos = u cos (u sin /) = x u sin y Y (x, y) α x φ x y i X

13 13 u sin R = If anle of pojection is chaned fom to = (90 ) then ane emains unchaned. hese anles ae called complementay anles of pojection. dr (i) Maximum ane: Fo ane to be maximum 0 d = d u sin = 0 d a pojectile will hae maximum ane when it is pojected at an anle of 45 o to the hoizontal and the maximum ane will be (u /). When the ane is maximum, the heiht H eached by the pojectile u sin u sin 45 u Rmax H = = = = 4 4 () Relation between hoizontal ane and maximum heiht: R = 4H cot If R = 4H then = tan 1 (1) o = 45. (9) Maximum heiht: It is the maximum heiht fom the point of pojection, a pojectile can each. So, by usin = u + as 0 = (u sin ) H (i) H max u sin H = u = (when sin = max = 1 i.e., = 90 o ) i.e., fo maximum heiht body should be pojected etically upwad. (10) Motion of a pojectile as obseed fom anothe pojectile is a staiht line..19 Hoizontal ojectile A body be pojected hoizontally fom a cetain heiht y etically aboe the ound with initial elocity u. If fiction is consideed to be absent, then thee is no othe hoizontal foce which can affect the hoizontal motion. he hoizontal elocity theefoe emains constant. (4) ime of fliht: If a body is pojected hoizontally fom a heiht h with elocity u and time taken by the body to each the ound is, then = (5) Hoizontal ane: Let R is the hoizontal distance taelled by the body h h R = u (6) If pojectiles A and B ae pojected hoizontally with diffeent initial elocity fom same heiht and thid paticle C is dopped fom same point then (i) All thee paticles will take equal time to each the ound. (ii) hei net elocity would be diffeent but all thee paticle possess same etical component of elocity. (iii) he tajectoy of pojectiles A and B will be staiht line w..t. paticle C. (7) If aious paticles thown with same initial elocity but indiffeent diection then

14 Kinematics / 14 (i) hey stike the ound with same speed at diffeent times iespectie of thei initial diection of elocities. (ii) ime would be least fo paticle which was thown etically downwad. (iii) ime would be maximum fo paticle A which was thown etically upwad..0 ojectile Motion on an Inclined lane. Let a paticle be pojected up with a speed u fom an inclined plane which makes an anle α with the hoizontal elocity of pojection makes an anle with the inclined plane. We hae taken efeence x-axis in the diection of plane. Hence the component of initial elocity paallel and pependicula to the plane ae equal to u cos Y and u sin espectiely i.e. u cos and u = u sin. u he component of alon the plane is sin α and X pependicula to the plane is cosα as shown in the a x= sin α t = fiue i.e. a = sin α and a = cos α. heefoe the paticle deceleates at a ate of sin α as it moes fom to. t =0 α a y= cos α usin u (1) ime of fliht: We know fo oblique pojectile motion = o we can say = a usin ime of fliht on an inclined plane = cos α () Maximum heiht: We know fo oblique pojectile motion u H = a Maximum heiht on an inclined plane (3) Hoizontal ane: R = u + α cos α sin cos( ) u sin H = cosα u sin H = o we can say π α (i) Maximum ane occus when = 4 (ii) he maximum ane alon the inclined plane when the pojectile is thown upwads u is ien by Rmax = (1 + sin α) (iii) he maximum ane alon the inclined plane when the pojectile is thown u downwads is ien by Rmax = (1 sin α) CIRCULAR MIN Cicula motion is anothe example of motion in two dimensions. o ceate cicula motion in a body it must be ien some initial elocity and a foce must then act on the body which is always diected at iht anles to instantaneous elocity.

15 15 Cicula motion can be classified into two types Unifom cicula motion and non-unifom cicula motion..1 Vaiables of Cicula Motion. (1) Displacement and distance: When paticle moes in a cicula path descibin an anle duin time t (as shown in the fiue) fom the position A to the position B, we see that the manitude of the position ecto (that is equal to the adius of the cicle) emains constant. i.e., 1 = = and the diection of the position ecto chanes fom time to time. (i) Displacement: he chane of position ecto o the displacement of the paticle fom position A to the position B is ien by efein the fiue. = 1 = sin (ii) Distance: he distanced coeed by the paticle duin the time t is ien as d = lenth of the ac AB = () Anula displacement (): he anle tuned by a body moin on a cicle fom some efeence line is called anula displacement. (i) Dimension = [M 0 L 0 0 ] (as = ac/adius). (ii) Units = Radian o Deee. It is some times also specified in tems of faction o multiple of eolution. (iii) πad = 360 = 1 Reolution (i) Anula displacement is a axial ecto quantity. Its diection depends upon the sense of otation of the object and can be ien by Riht Hand Rule; which states that if the cuatue of the fines of iht hand epesents the sense of otation of the object, then the thumb, held pependicula to the cuatue of the fines, epesents the diection of anula displacement ecto. () Relation between linea displacement and anula displacement s = o s = (3) Anula elocity (ω): Anula elocity of an object in cicula motion is defined as the time ate of chane of its anula displacement. (i) Anula elocity ω = anle taced time taken d = Lt = t 0 t dt d ω = dt (ii) Dimension: [M 0 L 0 1 ] (iii) Units: Radians pe second (ad.s 1 ) o Deee pe second. (i) Anula elocity is an axial ecto. Its diection is the same as that of. () Relation between anula elocity and linea elocity = ω 1 1 B 1 A B A S

16 Kinematics / 16 (i) Fo unifom cicula motion ω emains constant whee as fo non-unifom motion ω aies with espect to time. (4) Chane in elocity: We want to know the manitude and diection of the chane in elocity of the paticle which is pefomin unifom cicula motion as it moes fom A to B duin time t as shown in fiue. he chane in elocity ecto is ien as = 1 = sin Relation between linea elocity and anula elocity. In ecto fom = ω (5) ime peiod (): In cicula motion, the time peiod is defined as the time taken by the object to complete one eolution on its cicula path. (6) Fequency (n): In cicula motion, the fequency is defined as the numbe of eolutions completed by the object on its cicula path in a unit time. (i) Units: s 1 o hetz (Hz). (ii) Dimension: [M 0 L 0 1 ] Note: Relation between time peiod and fequency: = 1/n Relation between anula elocity, fequency and time peiod: π ω = = π n (7) Anula acceleation (α): Anula acceleation of an object in cicula motion is defined as the time ate of chane of its anula elocity. ω dω d (i) α = Lt = = t 0 t dt dt (ii) Units: ad. s (iii) Dimension: [M 0 L 0 ] (i) Relation between linea acceleation and anula acceleation a dω () Fo unifom cicula motion since ω is constant so α = = 0 dt (i) Fo non-unifom cicula motion α 0.. Centipetal Acceleation = α (1) Acceleation actin on the object undeoin unifom cicula motion is called centipetal acceleation. () It always acts on the object alon the adius towads the cente of the cicula path. 4π (3) Manitude of centipetal acceleation a= = ω = 4π n = (4) Diection of centipetal acceleation: It is always the same as that of..3 Centipetal Foce. Accodin to Newton's fist law of motion, whenee a body moes in a staiht line with unifom elocity, no foce is equied to maintain this elocity. But when a body moes alon a cicula F F F F

17 17 path with unifom speed, its diection chanes continuously i.e. elocity keeps on chanin on account of a chane in diection. Accodin to Newton's second law of motion, a chane in the diection of motion of the body can take place only if some extenal foce acts on the body. Due to inetia, at eey point of the cicula path; the body tends to moe alon the tanent to the cicula path at that point (in fiue). Since eey body has diectional inetia, a elocity cannot chane by itself and as such we hae to apply a foce. But this foce should be such that it chanes the diection of elocity and not its manitude. his is possible only if the foce acts pependicula to the diection of elocity. Because the elocity is alon the tanent, this foce must be alon the adius (because the adius of a cicle at any point is pependicula to the tanent at that point). Futhe, as this foce is to moe the body in a cicula path, it must acts towads the cente. his cente-seekin foce is called the centipetal foce. Hence, centipetal foce is that foce which is equied to moe a body in a cicula path with unifom speed. he foce acts on the body alon the adius and towads cente. m m4π (1) Fomulae fo centipetal foce: F = = mω = m4π n = () Centipetal foce in diffeent situation Situation A paticle tied to a stin and whiled in a hoizontal cicle. Vehicle takin a tun on a leel oad. A ehicle on a speed beake. Reolution of eath aound the sun Electon eolin aound the nucleus in an atom. A chaed paticle descibin a cicula path in a manetic field..4 Centifual Foce ension in the stin. Centipetal Foce Fictional foce exeted by the oad on the tyes. Weiht of the body o a component of weiht. Gaitational foce exeted by the sun. Coulomb attaction exeted by the potons in the nucleus. Manetic foce exeted by the aent that sets up the manetic field. It is an imainay foce due to incopoated effects of inetia. Centifual foce is a fictitious foce which has sinificance only in a otatin fame of efeence..5 Wok done by Centipetal Foce he wok done by centipetal foce is always zeo as it is pependicula to elocity and hence instantaneous displacement. Example: (i) When an electon eole aound the nucleus in hydoen atom in a paticula obit, it neithe absob no emit any eney means its eney emains constant. (ii) When a satellite established once in a obit aound the eath and it stats eolin with paticula speed, then no fuel is equied fo its cicula motion..6 Skiddin of Vehicle on a Leel Road. When a ehicle tuns on a cicula path it equies centipetal foce. µm mω

18 Kinematics / 18 If fiction poides this centipetal foce then ehicle can moe in cicula path safely if Fiction foce Requied centipetal foce m µ m safe µ his is the maximum speed by which ehicle can tun in a cicula path of adius, whee coefficient of fiction between the oad and tye is µ..7 Skiddin of bject on a Rotatin latfom. n a otatin platfom, to aoid the skiddin of an object (mass m) placed at a distance fom axis of otation, the centipetal foce should be poided by foce of fiction. Centipetal foce = Foce of fiction mω = µm ω max = ( µ / ), Hence maximum anula elocity of otation of the platfom is ( µ / ), so that object will not skid on it..8 Bendin of a Cyclist A cyclist poides himself the necessay centipetal foce by leanin inwad on a hoizontal tack, while oin ound a cue. Conside a cyclist of weiht m takin a tun of adius with elocity. In ode to poide the necessay centipetal foce, the cyclist leans thouh anle inwads as shown in fiue. m Rsin =..(i) and R cos = m..(ii) Diidin equation (i) by (ii), we hae tan =.. (iii) Note: Fo the same easons, an ice skate o an aeoplane has to bend inwads, while takin a tun..9 Bankin of a Road. R cos Fo ettin a centipetal foce cyclist bend towads the cente of cicula path but it is not possible in case of fou wheeles. heefoe, oute bed of the oad is aised so that a ehicle moin on it ets automatically inclined towads the cente. m / m R R sin

19 19 R R cos R sin l h m Fi. (A) x Fi. (B) o tan =... (iii) ω ω tan = =.... (i) [As = ω ] If l = width of the oad, h = heiht of the oute ede fom the ound leel then fom the fiue (B) h h tan = =.....() [since is ey small] x l Maximum safe speed on a banked fictional oad =.30 etunin of Vehicle. ( µ + tan ) 1 µ tan When a ca moes in a cicula path with speed moe than maximum speed then it oetuns and it s inne wheel leaes the ound fist Weiht of the ca = m Speed of the ca = Radius of the cicula path = Distance between the cente of wheels of the ca = a Heiht of the cente of aity (G) of the ca fom the oad leel = h a he maximum speed of a ca without oetunin on a flat oad is ien by =. h.31 Non-Unifom Cicula Motion. If the speed of the paticle in a hoizontal cicula motion chanes with espect to time, then its motion is said to be non-unifom cicula motion. usin υ = ω..(i) the esultant acceleation of the paticle at has two component acceleations (1) anential acceleation: a = α t It acts alon the tanent to the cicula path at in the plane of cicula path. () Centipetal (Radial) acceleation: ac = ω It is also called centipetal acceleation of the paticle at. It acts alon the adius of the paticle at. he manitude of centipetal acceleation is ien by a = ω υ =ωυ sin90 = ωυ=ω( ω ) = ω = υ / c

20 Kinematics / 0 Hee at oens the manitude of while a c its diection of motion..3 Equations of Cicula Motion. Fo acceleated motion ω = ω1 + αt 1 = ω 1t + α t Fo etaded motion ω = ω1 + αt 1 = ω1t α t ω = ω + α 1 1 ω = ω α α α n = ω 1 + (n 1) n = ω1 (n 1).33 Motion in Vetical Cicle. his is an example of non-unifom cicula motion. In this motion body is unde the influence of aity of eath. (1) Velocity at any point on etical loop: If u is the initial elocity impated to body at lowest point then. Velocity of body at heiht h is ien by u h u l = = (1 cos ) whee l in the lenth of the stin () ension at any point on etical loop: ension at eneal point, (3) Vaious conditions fo etical motion: Velocity at lowest point ua ua Condition m = mcos + l > 5l ension in the stin will not be zeo at any of the point and body will continue the cicula motion. = 5 l, ension at hihest point C will be zeo and body will just complete the cicle. l < ua < 5 l, aticle will not follow cicula motion. ension in stin become zeo somewhee between points B and C wheeas elocity emain positie. aticle leaes cicula path and follow paabolic tajectoy. ua ua Whee ω1 = Initial anula elocity of paticle ω = Final anula elocity of paticle α = Anula acceleation of paticle = Anle coeed by the paticle in time t n = Anle coeed by the paticle in n th second = l Both elocity and tension in the stin becomes zeo between A and B and paticle will oscillate alon semi-cicula path. < l elocity of paticle becomes zeo between A and B but tension will not be zeo and the paticle will oscillate about the point A. D l h C A B u

21 1 (6) Vaious quantities fo a citical condition in a etical loop at diffeent positions: Quantity oint oint B oint C oint D oint A Linea elocity () 5l 3l l 3l l (3 + cos ) Anula elocity (ω) (3 + cos ) l l l l l 6 m 3 m 0 3 m 3m (1 + cos ) ension in Stin () Kinetic Eney (KE) 5 ml 3 ml 1 ml 3 otential Eney ml mu 5 m = l 0 0 ml ml ml Ml(1 cos ) (E) otal Eney (E) 5 ml 5 ml 5 ml 5 ml 5 ml (7) Motion of a block on fictionless hemisphee: A small block of mass m slides down fom the top of a fictionless hemisphee of adius. he component of the foce of aity (m cos ) poides equied centipetal foce but at point B it's cicula motion ceases and the block lose contact with the suface of the sphee. A B m h ( h) Fo point B, by equatin the foces, by law of conseation of eney m mcos =...(i) otal eney at point A = otal eney at point B the block lose contact at the heiht of 3 fom the ound.