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

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1 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, which means that diffeent pats of the object wee moing along the same path and the motion was the same as if the object had no size. Een though this type of motion is ey common, this is not the only type of motion existing in natue. Anothe well known case of motion is otation o cicula motion. The fist example which comes in mind is the motion of planets. They ae otating aound thei own axes of otation as well as taeling along almost cicula paths aound the Sun. 1. Unifom cicula motion We shall stat fom the simplest example of the cicula motion: unifom cicula motion, when a paticle taels along the cicula ac at constant speed. Notice that we used the wod "speed" not "elocity". This is because elocity is a ecto and if we say constant elocity, this would mean that paticle moes not just at constant ate but also in the same diection all the time. It is not tue fo a cicula motion. Indeed een if the paticle's speed (magnitude of it elocity) is constant, the diection of its motion is changing all the time. Since its elocity is changing, it should also hae some acceleation. Let us ty to find this acceleation duing the cicula motion. We shall conside a paticle which moes aound a cicle with elocity of constant magnitude. The cicle has adius. The position of the paticle is detemined by its coodinates, x cos, and y sin with the oigin of coodinate system located at the cente of the cicle (see the pictue). As we aleady know, the elocity ecto always has the diection tangential to the path of motion. In this case it is tangential to the cicle and pependicula to the adius of this cicle, making the same angle,, with the etical diection as the adius-ecto makes with axis x. So, one can esole ecto into components as To find acceleation, we hae to use its definition as the fist deiatie fom elocity with espect to time

2 Accoding to this, the diection of the acceleation ecto, a, should be along the adius ecto, but the negatie sign shows that it has the opposite diection. This means that it points towads the cente of the cicle. This is why we call this acceleation to be centipetal acceleation. The common mistake is to say that acceleation is diected away fom the cente of the cicle, which is wong. Indeed, in ode to be able to make a tun, the paticle should change its diection of motion towads the cente of the ac, so its acceleation should be diected in a same way. The magnitude of this acceleation is F x y H G I K J F H G I K J b g b g. (3.3.1) a a a cos sin cos sin When this paticle taels aound the cicle, it makes one complete eolution aound the cicumfeence fo time known as the peiod of eolution T. (3.3.)

3 Example What is the minimum elocity necessay to put a satellite on the Eath's obit and what is the peiod of otation fo this satellite? Since we ae talking about the minimum elocity necessay, we can make an assumption that this satellite otates extemely close to the Eath's suface, so we can appoximate the adius of its obit by the adius of the Eath R E m. We shall also neglect ai esistance. Then acceleation of this satellite will be just acceleation due to gaity, g 9. 8m s. It points towads the cente of the Eath, so it is centipetal acceleation and as a esult we hae g R E. This gies gr 9. 8m s m 7. 9km s, which the minimum speed E 6 needed fo the satellite to leae the Eath's suface without falling back. The peiod of otation fo this satellite will be RE m T 5060s 84 min m s 6 Note that this elocity of the satellite is popotional to the squae oot of the obit's adius, but this is only tue nea the eath's suface, since gaitational acceleation, g, becomes smalle futhe away fom the eath. We shall etun to this question, when we will study gaitational foce.. Rotation Up till this point we hae discussed motion of the point-like objects along the diffeent paths, including motion aound the cicle. But not all the objects behae like paticles. Fo instance, planets, including ou Eath, ae not just moing along almost cicula paths aound the Sun, but they ae also otating aound thei own axes. This is also tue fo diffeent wheels and pulleys. Fo example, when taeling along the oad, ca's o bicycle's wheels paticipate in two types of motion: tanslational motion with espect to the oad and otational motion aound thei own otational axes. We hae aleady quite succeeded in desciption of tanslational motion, now we hae to conside otation. Let us stat fom the pue otation of the igid body, which otates with all its pats locked togethe without any changes of its shape o position of its otational axis. To descibe such a motion, we hae to pick up the appopiate set of coodinates. Simila to

4 the case of tanslational motion, when the best choice was to conside the change of position in the diection of motion, now we can conside the change of angula position. To do so, we hae to define a efeence line in the body pependicula to the axis of otation. Then we can count the angula position as an angle, which this line makes with espect to zeo angula position, fo instance with espect to axis x fixed in the space. If we measue the angle in adians the length of the cicula ac s made by the point of the igid body fixed at distance fom the otational axis will be s. (3.3.3) In paticula, when this point makes one complete eolution, the angle becomes and we hae s fo the length of the cicumfeence. The length of the cicula ac, s, will be diffeent fo diffeent points of the body at diffeent adii fom the axis of otation. At the same time, angle is a pefect choice of the aiable to descibe otation of the body, since it does not depend on paticula distance fom the otational axis. To study tanslational motion we wee tying to obtain the dependence, t b g, of position as function of time. In a same way, now we need to know the angula position, btg, as function of time. When body is otating changing its angula position fom 1 to duing the time inteal t t t, we can define the angula displacement as 1 1. (3.3.4) The angula displacement in counteclockwise diection consideed to be positie, in clockwise diection negatie. One can define the aeage angula elocity fo a gien time inteal t as ag 1 t t t. (3.3.5) 1 The instantaneous angula elocity at cetain time instant is (3.3.6) which is the fist deiatie fom angula position with espect to time. These definitions of angula displacement and angula elocity ae going to be alid not only fo a otating body but fo a paticle moing aound the cicle too. As it can be seen fom its definition, the SI unit fo the angula elocity is ad/s, while othe units such as e/s (eolutions pe second) and pm (eolution pe minute) ae also used. If a body otates counteclockwise,

5 it has positie angula elocity. If it otates clockwise, it has negatie angula elocity. The magnitude of angula elocity is called the angula speed. If a body otates at non-constant ate, one can define its angula acceleation. The aeage angula acceleation fo the time inteal t is ag 1 t t t. (3.3.7) 1 If we shink this time inteal to zeo, the instantaneous angula acceleation can be intoduced as the fist deiatie of the angula elocity with espect to time o the second deiatie of the angula position with espect to time (3.3.8) The SI unit fo the angula acceleation is ad s. This definition is also alid fo a paticle moing aound the cicle. In ou example at the beginning of this lectue, the paticle was moing with constant speed. In that case the angula acceleation was zeo een though the centipetal acceleation was not zeo. As we saw in the case of the motion along the staight line, een though displacement, elocity and acceleation ae ectos we only needed to indicate thei signs to detemine diection in one dimension. The simila situation occus fo otation aound one fixed axis, when we call angula elocity to be positie if otating counteclockwise and negatie if otating in the clockwise diection. Howee, angula quantities ae also ectos. The diection of the angula elocity ecto is defined accoding to the ight hand ule: Cul you ight hand about object's axis of otation with you finges pointing in the diection of otation then you extended thumb will point in the diection of the angula elocity ecto. The same is tue fo the angula acceleation. Howee, in contast to tanslational motion, angula displacement is not a ecto, because it does not obey commutatie law. 3. Angula and Linea Vaiables In the case of tanslational motion along the staight line we hae outlined the special case when acceleation was constant. One can conside the same special case fo otational motion. Moeoe, we do not hae to deie all of the equations again. In the

6 case of the motion with constant angula acceleation,, we can just eplace the egula linea quantities with angula quantities in ou old equation set to obtain const, t, 0 t 0 0t. These ae fundamental equations. Eliminating time o angula acceleation gies 0, t. b b g 0 g (3.3.9) (3.3.10) We also know that simple elation exists between the absolute alue of displacement and angula displacement fo any point of the igid body: s (3.3.11) The simila elations can be established between othe otational and tanslational aiables. Indeed taking the fist deiatie of the equation , we hae (3.3.1) If the angula elocity is constant, all the points of the igid body ae otating with the same angula elocity, but linea elocity becomes lage fo lage adii. The peiod of otation then becomes T (3.3.13) Acceleation at any point can be calculated as (3.3.14) Hee we hae used lette, a t, to epesent acceleation, instead of just a, because this a t is the tangential component of acceleation, which is esponsible fo the change of the magnitude of acceleation. If we hae otation with constant speed, so a t is zeo, the only component of acceleation is the one due to change of elocity s diection. That

7 component of acceleation is nomal to the path and diected towads the cente of the cicle. In geneal it is called the nomal component of acceleation b g. (3.4.15) an The net acceleation of the point of the otating body is t n b g d i. (3.4.16) a a a Example Find constant angula acceleation of the wheel, if it is known that afte the.0 seconds of otation with this angula acceleation the net acceleation ecto of a point on the wheel's im makes angle 60 degees with its linea elocity. Assume that wheel's motion has stated fom est. The net acceleation ecto has two components, the nomal o centipetal component, which is an and the tangential component, which is at. Accoding to the poblem this ecto makes angle 60 degees with the linea elocity, which in the tangential diection to the wheel, so tan60 o an a t On the othe hand is the angula elocity of the wheel afte.0 seconds of otation which has stated fom est, so 0 t t, and we hae b g o t tan 60 t, o tan 60 tan 60 ad t. 0s s b g o

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