Velocity and Accceleration in Different Coordinate system

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1 Velocity & cceleration in different coordinate system Chapter Velocity and ccceleration in Different Coordinate system In physics basic laws are first introdced for a point partile and then laws are etended to system of particles or continos bodies. Therefore, we also bein the discssion with point particle and later on we will stdy collection of particles or riid body. To write eqations oernin the dynamics of a aprticle we need its position ectors, elocity, acceleration etc. Therefore we first introdce these elementaty concepts. Position ector: It is a ector directed from some point to location of particle. In the fire shown r is position ector of particle P with respect to point. If we specify the coordinate of particle then position ector can be epressed in terms of coordinates and nit ectors sed in that coordinate system. In cartesian coordinate system: Coordinates of particle are written as (, y, z) and nit ectors alon, y, z aes are ˆ, yˆ, and z ˆ respectiely. Therefoer, from fire, z =, = y, P = z P ^z (, y, z) and P P r r P ˆ yyˆ zzˆ y ^y r ˆ yyˆ zzˆ ^ Unit ectors are taken in the directions in which coordinates increase. In cylindrical system coordinates of particle are written as s,, z and nit ectors alon the increasin direction of coordinates are sˆ, ˆ, zˆ. s is perpendiclar distance of particle from z-ais, is its anlar position with respect to -ais and z is its distance aboe -y plane. z X 90º 90º s r s 90 z^ z^ ^ Y s^ s^ r P

2 Velocity & cceleration in different coordinate system Therefore, from fire s,, P z and P P r ssˆ zzˆ relation with cartesian coordinates s cos, y s sin, z z and sˆ cos ˆ sin yˆ, ˆ sin ˆ cos yˆ, z zˆ In spherical polar coordinates system, coordinates of particle are written as r,, and nit ector in increasin direction of coordinates are rˆ, ˆ and ˆ. r is the distance of particle from oriin, and are anlar position with respect to z and aes respectiely. From fire P r, CP, and P rrˆ z r rrˆ ^ C CP r sin, P r cos s P r sin r sin.sin and cos r sin cos Relation with cartesian coordinate, r sin.cos, y r sin sin, z r cos rˆ sin cosˆ sin yˆ cos zˆ nd ˆ cos cos ˆ sin y ˆ sin z ˆ ˆ sin ˆ cos yˆ If motion of a particle is confined in one plane then only two coordinates are reqired to describe its position. We can either se cartesian coordinates (, y) or plane polar coordinates s,. Ths if a particle is moin on a plane then its position ector can be written as Y ^ s^ r s r ˆ yyˆ r, r ss ˆ in (plane polar coordinate) Plane polar coordinates s, are the same coordinates which are sed in cylindrical coordinates system. Notice that, ˆ, yˆ and z ˆ hae a fied direction as they are alon the, y and z aes, whereas rˆ, sˆ, ˆ, ˆ etc do not hae fied directions. Therefore, ˆ, yˆ, z ˆ are constant nit ectors bt ectors. dˆ dyˆ dzˆ Ths, 0, 0, 0 and Deriatie of nit ectors rˆ, ˆ, sˆ, ˆ X X drˆ dsˆ dˆ d 0, 0, 0, 0 90º r r z^ Y rˆ, sˆ, ˆ, ˆ are not constant nit can easily be fond by sin their relation with ˆ, ˆ, ˆ y z.

3 Velocity & cceleration in different coordinate system 3 For eample: In plane polar or cylindrical coordinates, sˆ cos ˆ sin yˆ and ˆ sin ˆ cos yˆ ˆ ds sin d ˆ cos d yˆ d y sin ˆ cos ˆ ˆ and d ˆ cos d ˆ sin d yˆ d cos ˆ sin yˆ sˆ dsˆ ˆ ˆ d, sˆ Velocity: erae elocity is defined as, a total displacement chane in position ector total time taken total time taken r r t t Instantaneos elocity (elocity at any instant of time) is defined as time deriatie of position ector. dr Instantaneos elocity,. y epressin r in different coordinate system. In cartesian coordinate system: r ˆ yyˆ zzˆ d ˆ yy ˆ zz ˆ Therefore, ˆ yyˆ zzˆ r, ˆ yˆ zˆ y z y z d dy dz ˆ yˆ zˆ d is component of elocity in direction. dy y is component of elocity in y direction dz z is component of elocity in z direction In plane coordinate system: r ssˆ d ds ˆ ss sˆ s ds Therefore, ˆ

4 4 Velocity & cceleration in different coordinate system r, s ˆ ˆ ˆ ˆ ss s ds sˆ ˆ s ds s is component elocity in ŝ direction and it is called radial elocity d s s is component elocity in ˆ direction and it is called transerse elocity. In cylindrical coordinate system: r ssˆ zzˆ dr d ssˆ zzˆ Therefore, ˆ ds sˆ s ds dz zˆ ˆ ˆ ˆ ˆ ˆ ss s zz ds cceleration : erae acceleration is defined as aa t t Instantaneos acceleration is defined as time deriatie of elocity ector. d a y epressin indifferent coordinate system. We can write acceleration of a particle in different coordinate system. In cartesian coordinate: ˆ yy ˆ zz ˆ d d a ˆ yy ˆ zz ˆ a ˆ yyˆ zzˆ a a ˆ a yˆ a zˆ Therefore, or, y z d d a d d ay y y y is component of acceleration alon -direction. is component of acceleration alon y-direction. d dz az z z is component of acceleration alon z-direction. Since, a be zero. d, therefore, if elocity alon -direction is constant then acceleration alon -direction mst

5 Velocity & cceleration in different coordinate system In Plane polar coordinate : ss ˆ s ˆ d d Therefore, a ss ˆ s ˆ dsˆ ˆ ˆ ˆ ˆ d ss s s s s r, s ˆ ˆ ˆ ˆ ssˆ s s s s s ˆ a s s s s s ˆ a a s a ˆ ˆ s is component of acceleration alon ŝ and its called radial acceleration. Clearly, it is not eqal to a s s time deriatie of radial component of elocity Therefore, if s a s s constant s s., then a s may not be zero. is component of acceleration in ˆ direction. It is called transerse acceleration. Clearly it is also not eqal to time deriatie of radial component of elocity s d i.e. a Therefore, if constant then a may not be zero. In cylindrical coordinate system: ss ˆ s ˆ zz ˆ First two terms are same as in plane polar coordinate. d Therefore, a s s sˆ s s ˆ zzˆ How to choose a coordinate system: Generally speakin, dynamics of a particle can be stdied sin any coordinate system. Howeer jdicios selection of coordinate system makes task easy. ne dimensional cases is simple becase we jst need one coordinate which we can take either or y or z. Howeer in two dimensional cases one has to be contios. In most of the two dimensional problems if acceleration of the particle is directed towards a point then plane polar coordinate is coneninet to se. Howeer, if accleration is not directed towards a point then cartesian coordinate is riht choice. For eample : projectile motion in which particle is projected with a small speed at some anle with horizontal. It moes with constant acceleration which is directed downward. In this case acceleration is not directed towards a point. Therefore, we can coneniently se cartesian coordinates. a 5 a a a In planetary motion, acceleration of planet is always directed towards centre of the sn. Therefore in this case plane polar coordinate makes calclation simple. If a particle is attached to a point with a strin or sprin then also enerally polar coordinate is helpfll. For eample, in case of simple pendlm.

6 6 Velocity & cceleration in different coordinate system Formlae of kinematics (niformly accelerated motions): When a constant force acts on a particle its acceleration is constant. If, be initial and final elocity of a particle alon -direction and a be its acceleration alon direction then. at a if a constant t at where is displacement alon -direction and t is time taken. We can write similar relations for y and z direction also. Note: If acceleration of particle is not constant then we cannot se formlae of kinematics. In that case we start with either definition of elocity or definition of acceleration i.e. d or a d F d d d d we may also hae to se a and. as the case may be m d d Projectile motion: If be the projection speed and be anle of projection then eqation of path of projectile is, y tan cos Rane, sin R Maimm heiht, sin H sin Total time of fliht, T Coordinate of projectile at any time t < T. cos t, y sin t t Note: boe reslts for projectile are applicable only if projectile is thrown from rond and it finally lands on rond. nd projection speed is small so that projectile does not o too hih. If projection speed is not small then heiht will be lare and in that case we need to consider ariation of acceleration de to raity. y H R

7 Velocity & cceleration in different coordinate system 7 SLVED PRLEMS. particle moes alon a circle of radis R = 50 cm so that its radis ector r relatie to point (fire (a)) rotates with the constant anlar elocity 0.40 rad/sec. Find the modls of the elocity of the particle and modls and direction of its total acceleration. r C R Y r R C X (a) (b) r R Soln. Consider X and Y aes as shown in (fire (b)). Usin sine law in trianle C, we et sin ( ) sin or r R sin cos sin r R cos Now r cos r sin R cos R cos sin r i j i j dr d d Now 4R cos sin i R cos j R sin i j Frther R d d d a 4R cos i 4R sin j 4R cos i 4R sin j a 4 R.. fter bein hit, a olf ball reaches a maimm heiht of 60m with speed of 0m/s. Riht after bein hit the speed of the ball is (take = 0 m/s ) [GP-008] (a) 00m/s (b) 80m/s (c) 60m/s (d) 40m/s Soln. (d) If be the speed jst after bein hit at anle then cos 0... (i) sin cos = 0 sin = (ii) on sqarin (i) and addin to (ii) we et

8 8 Velocity & cceleration in different coordinate system 3. particle starts moin with acceleration a where and are constants and is its instantaneos Soln: speed. Find elocity of particle as a fnction of time and also find its terminal elocity. cceleration is ariable therefore we start with definition of acceleration d a or ln t as t, d or or, d 0 0 t e t ( e t ) = constant. This is terminal elocity. 4. projectile is thrown at an anle with horizontal with initial speed after what time the projectile s elocity will be perpendiclar to its initial direction. Soln. Sppose elocity after time t becomes perpendiclar to initial elocity. Velocity after time t is ( t) i ˆ yj ˆ cos iˆ ( sin t) ˆj Initial elocity (cos iˆ sin ˆj ) is to, therefore,. 0 cos sin t sin 0 0 t sin 5. ball is dropped from a heiht H. It bonces back p to a heiht e times after hittin the rond. If e<, after what time the ball will finally come to rest. Soln. all moes nder the effect of raity. Therefore manitde of acceleration of ball drin pward or downward moement remains constant. For first bonce y H, 0, a y yt ayt H 0 t y y t H =H H =eh H 3=e H H 3 H 4=e H H Therefore time of fall before first bonce t Under raity, time for oin p a certain distance is eqal to time for oin down. Therefore time interal between first bonce and second bonce is t H eh 3 e H e H Similarly t3, t4,... Therefore, total time elapsed before ball stops is T t t t3... H / / 3/ e e e... / H e H e / e e

9 Velocity & cceleration in different coordinate system 9 6. particle is projected with elocity ( h ) so that it jst clears two walls of eqal heiht h which are at a distance h from each other. Show that the time of passin between the walls is ( h/ ). Soln. The sitation is shown in fire. Y h h h D h E Sppose the particle is at a heiht h after time t. Then, h h sin t t or t 4 h sin t h 0... (i) This eqation ies the two ales of time at which the heiht of the particle is h. Let these times be t and t respectiely. Then, ( t t ) ( t t ) 4t t From eqation (i) 4 ( h) sin h t t and t t 6h sin 8h 8h ( t t) [ sin ] C X 8h or ( t t) [ sin ] /... (ii) Drin this time, the horizontal distance moed is h. The horizontal elocity bein ( h) cos. Usin s t, we hae h ( h) cos t or h ( h) cos (8 h/ [ sin ] Sqarin 8 cos ( cos ) 4 or 6 cos 8 cos 0, or (4 cos ) 0. / Sbstittin these ales in eqation (ii) we hae cos and sin.... (iii) 3 ( t t ) [8 h / ] 4 or t t ( h/ ). /

10 0 Velocity & cceleration in different coordinate system 7. Two particles are projected from a point at the same instant with elocities whose horizontal and ertical Soln. components are, and, respectiely. Proe that the interal between their passin throh the other ( ) common point of their path is. ( ) The sitation is shown in fire. Y N P X Let the two particles pass throh the common point P. ain let, t and t be the time taken by the particles to o to P from. lon X-ais, we hae t t In the ertical direction... (i) t t PN t t... (ii) From eqation (ii), we hae t t / ( t t )... (iii) t t from eqation (i),. t t t ddition one to both sides, we hae. t ( ) ( t t) t... (i) From eqation (iii), we et ( t t) ( t t) ( t t) ( t t) t t t 8 ( ) ( ) ( t t) ( t t) ( ) ( ) or ( t t). ( ) 8. Particles P and Q of mass 0 m and 40 m respectiely are simltaneosly projected from points and on the rond. The initial elocities of P and Q make 45º and 35º anles respectiely with the horizontal as shown in the fire. Each particle has an initial speed of 49 m/sec. The separation is 45 m. oth particles trael in the same ertical plane and ndero a collision. fter the collision P retraces its path. Determine the position of Q when it hits the rond. How mch time after the collision does the particle Q take to reach the rond. Take = 9.8 m/sec.

11 Velocity & cceleration in different coordinate system P Q 35º 45º 45º Soln. The horizontal elocity of either particle is 49 cos 45º 49 m/sec. Initially, the ertical component of elocity of either particle 49 sin 45º 49 m/sec. The horizontal distance traelled by each particle at the time of collision = 45/ metre. The ertical component of the elocity of each particle at the time of collision is t the time of collision, the elocity of each particle is horizontal and is eqal to 49 Let p and q be the elocities after collision, then s P retraces its path, p p 0.04 q. 49. m/sec ( ) q The particle falls ertically downward from the point of collision, i.e., at a distance of.5 metre from. Let t be the time taken, then t. Since collision does not after the ertical component of elocities t 3.53 sec Two shots are projected from a n at the top of a hill with the same elocity at anles of projection and respectiely. If the shots strike the horizontal rond throh the foot of the hill at the same point, show that the heiht h of the hill aboe the plane is ien by Soln. h ( tan tan ). (tan tan ) Let R be the horizontal rane for both. Takin the point of projection as oriin, the point strck is (R, h). This point satisfies the eqations of both the trajectories. R h R tan cos... (i) R and h R tan cos From these eqations, we hae R R R tan R tan cos cos... (ii)

12 Velocity & cceleration in different coordinate system R R or R (tan tan ) cos cos / (tan tan ) R (sec sec ) or / (tan tan ) R[( tan ) ( tan )] or / R (tan tan ).... (iii) R (tan tan ) Pttin the ale of R in eqation (i), we hae R sec h R tan tan cos (tan tan ) tan tan sec tan tan tan ( tan tan ). (tan tan ) (tan tan ) 0. particle is moin alon a ertical circle of radis r 0 m with a constant speed 3.4 m/s as shown in fire below. Straiht line C is horizontal and passes throh the center of the circle. shell is fired from point at the instant when the particle is at C. If distance is 0 3 m and the shell collide with the particle at, then proe (n ) tan, 3 where n is an inteer. Frther show that smallest ale of is 30º. 0 3 m r 0m = C Soln. s at the time of firin of the shell, the particle was at C and the shell collides with it at, therefore the nmber of reoltions completed by the particle is odd mltiple of half i.e., ( )/, where n is an inteer. Let T be the time period of the particle, then r T 4 second n If t be the time of fliht of the shell, then t = time of [(n )/] reoltions of the particle (n ) 4 (n ) second sin for a projectile, the time of fliht is ien by t. Hence, sin (n )... (i) sin The rane of the projectile is ien by R.

13 Velocity & cceleration in different coordinate system 3 Hence, sin (ii) (n ) From eqations (i) and (ii), we et tan. 3 For to be smallest, n, so min 30º.. n object is kept fied at the point 3m and y.5 m on a plank P raised aboe the rond. t time t 0 the plank starts moin alon the direction with an acceleration.5 m/s. t the same instant a stone is projected from the oriin with a elocity as shown. stationary person on the rond obseres the stone hittin the object drin its downward motion at an anle of 45º to the horizontal. ll the motions are in the yplane. Find and the time after which the stone hits the object. Take 0 m/s. y.5 m P 3.0 m Soln. See fire. Y 3.0 m P.5 Let be the anle of projectile. Frther, sppose the stone hits the object at time t, then ( cos ) t t... (i) and ( sin ) t t.5... (ii) s the stone hits the object, moin alon 45º with the horizontal drin downward motion, we hae y sin t tan ( 45º ) cos sin t or cos or sin t cos... (iii) P 45º X

14 4 Velocity & cceleration in different coordinate system 3.0 From eqation (i), cos 0.75t t.5 From eqation (ii), sin t t Sbstittin these ales in eqation (iii), we et t 0t 0.75t t t 4.5 t From eqation (i) and (ii), we et 4.5t or t or t sec. cos (i) and sin () Diidin eqation () eqation (i), we et tan or tan Sqarin and addin eqation (i) and (), we et (3.75) (6.5) (4.06) (39.06) 53. or 7.9 m/s.. rotation disc (fire below) moes in the positie direction of the -ais. Find the y( ) describin the positie of the instantaneos ais of rotation, if at the initial moment the ais C of the disc was located at the point after which it moed (a) with a constant elocity, while the disc started rotatin conter clockwise with a constant anlar acceleration (the initial anlar elocity is eqal to zero); (b) with a constant acceleration a (and the zero initial elocity), the disc rotates conter clockwise with a constant anlar elocity. Soln. (a) When the disc rotates with constant anlar elocity and conter clockwise anlar acceleration : We hae the relation 0,. t t t Now consider any point P of the disc at a distance y from the -ais; i.e. r y j. n ais passin throh P and parallel to the ais of rotation. Now the linear elocity of point P. p r and k k y( j) y i; that is, the point P has a elocity alon the -ais which is the same as the elocity of any point of the disc =. Hence, y : (eqation of a hyperbola) (b) When the disc rotates with constant anlar elocity and constant acceleration a : In this case a t at 0 a t t

15 Velocity & cceleration in different coordinate system 5 Hence, a t a ain r k y ( j), p a Therefore, y or a y or y, a parabola]. 3. balloon rises from rest on the rond with constant acceleration /8. stone is dropped when the balloon has risen to a heiht H meter. Show that the time taken by the stone to reach the rond is ien by ( H / ). Soln. The elocity of the balloon when it has risen to a heiht H is ien by 0 H takin pward 8 direction positie. ( H ) ( H ) m/sec. 4 This will also be the elocity of the stone in pward direction when it is dropped. ( H ) Takin pward direction position for the stone, H t t t ( and h H ) or t ( H ) t H 0 ( H ) {( H ) (8 H )} ( H ) 3 ( H ) H t takin positie ale particle is projected ertically pwards. Proe that it will be at 3/4 of its reatest heiht at time which are in the ratio : 3. Soln. We know that the reatest heiht attained / (where is the initial elocity). Let t be the time when the particle is at a heiht Usin the formla s t t, we hae t t t t Solin for t, we hae t. Takin neatie sin, t. 3 Takin only positie sin, t. t ( / ) t (3 / ) : 3.

16 6 Velocity & cceleration in different coordinate system 5. Two particles, and, moe with constant elocities and alon two mtally perpendiclar straiht lines towards the intersection point. t the moment t 0 the particles were located at the distances l and l from the point. How soon will the distance the particles become the smallest? What is it eqal to? t Z l l t t l l t Soln. t a time t, the positions of the particle and are separately, ( l t, ) and (0, l t). Hence the distance between them at this instant. Z ( l t) ( l t) To be minimm at some instant t t, we se the condition dz ( l t) ( ) ( l t) ( ) l t l t 0 0 ( ) ( ) l or ( ) t l l or t t l l l l l Hence Zmin l l. ( ) l l 6. From point located on a hihway, one has to et by as soon as possible to point located in the field at a distance l from the hihway. It is known that the moes in the field times slower than on the hihway. t what distance from the point D one mst trn off the hihway? X C D l Soln. Let the car trn off the road at a distance from D. Hence the total time of trael by car from to. T X l /

17 Velocity & cceleration in different coordinate system For this to be minimm : dt 0 0 d l l 7 l l l or. 7. Two points are moin with niform elocities and alon the perpendiclar aes, X and Y. The motion is directed towards, the oriin. When t 0, they are at a distance a and b respectiely from. Calclate the a b anlar elocity of the line joinin them at time t. Show that it is reatest, when t. Soln. See fire. t t 0, the particles are at and respectiely. Y Q b a P X Let at time t, the particles are at P and Q respectiely. From fire, P a t... (i) and Q b t... (ii) Let be the anle which the line PQ makes with the direction X at time t. From QP, Q b t b t tan ( ) or tan P a t a t b t b t or tan or tan t a t a... (iii) d ( t a) ( ) ( b t) Differentiatin eqation (iii), we et b t ( t a) t a d a b Simiplifyin it, we et ( a t) ( b t) becase nmerator is constant. Let Z ( a t) ( b t) dz ( a t) ( ) ( b t) ( ) 0 a b Solin it for t, we et t., will be maimm when denominator is minimm

18 8 Velocity & cceleration in different coordinate system It can be seen that d Z a b is positie. Hence is maimm, when t. 8. particle of mass m starts from rest moin down the inclined plane and rises p to point C on the inclined plane C. ssmin no losses of enery, the time period of the to and for motion of the particle is... h h h Soln. The time taken t in moin from to, is ien by 0 ( sin ) ( s ). sin sin h Therefore, t sin h Similarly, the time taken, t from C to is ien by t sin. h Time period, T ( t t). sin sin 9. particle moes in the plane y with constant acceleration a directed alon the neatie y-ais. The eqation of motion of the particle has the form y p q where p and q are positie constants. Find the elocity of the particle at the oriin of coordinates. Soln. Gien that y p q dy d d d y d d d d p q and p q q or a q q d d y 0 (no acceleration alon -ais), and a. a a or q q dy Frther, 0 d p p or y a y p q a ap a ( p ) Now, ( y ) or. q q q

Motion in Two Dimension (Projectile Motion)

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