Chapter 12: Kinematics of a Particle 12.8 CURVILINEAR MOTION: CYLINDRICAL COMPONENTS. u of the polar coordinate system are also shown in

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1 ME 01 DYNAMICS Chapte 1: Kinematics of a Paticle Chapte 1 Kinematics of a Paticle A. Bazone 1.8 CURVILINEAR MOTION: CYLINDRICAL COMPONENTS Pola Coodinates Pola coodinates ae paticlaly sitable fo solving poblems fo which data egading the angla motion of the adial coodinate is given to descibe the paticle s motion. Fige 1 shows the pola coodinates and that specify the position of the paticle P that is moving in the plane. The oigin is established at a xy fixed point, and the adial line diected to the paticle. The tansvese coodinate is meased conteclockwise fom a fixed efeence line to the adial line. In plana motion, the pola coodinate is eqal to the magnitde of the position vecto of the paticle. j y i Fig. 1 P x Deivative of the Unit Vectos The nit vectos and is diected along the adial line, pointing away fom O ;, in the diection of inceasing. Fom Fig.1, it is clea that the nit vectos and Fig. 1. The vecto pependicla to of the pola coodinate system ae also shown in is will otate as the paticle moves. Theefoe, and ae the base vectos of a otating efeence fame, simila to the path ( n t) coodinate system. The diffeence between the n t coodinates depend on the path and diection two coodinate systems is that ( ) of the paticle, wheeas pola coodinates ae detemined by the position of the paticle. Conseqently, these base vectos possess nonzeo deivatives, even thogh thei magnitdes ae constant (eqal to one). The time deivative of the nit base 1/11

2 ME 01 DYNAMICS Chapte 1: Kinematics of a Paticle vectos can be detemined by fist elating the vectos to the xy coodinate system. Fom Fig. 1, = cos i + sin j = sin i + cos j i j Diffeentiating with espect to time while noting that d dt = d dt = 0 (the xy coodinate system is fixed) yields d d ( ) dt = = ( sin i + cos j) ( ) dt = = ( cos i sin j) (1) () Compaing Eqs (1) and (), one can find =, = (3) The tem is called the angla velocity of the adial line. The base vectos and thei deivatives ae shown in Fig.. Notice that and, espectively. and ae pependicla to y = y j i P x j = i P x Fig. Unit vectos and thei deivatives /11

3 ME 01 DYNAMICS Chapte 1: Kinematics of a Paticle Velocity and Acceleation Vectos The position vecto of the paticle can be witten in pola coodinates as = (4) Since the velocity vecto is, by definition, v = d dt, we have d d d d v = = ( ) = + = + dt dt dt dt Sbstitting fo d( ) dt = = fom Eq.(3) gives v = v + v = + (5) whee v =, and v = (6) v and v ae called The components the adial and tansvese components of the velocity, espectively. These components ae mtally pependicla and the magnitde of the velocity o speed is given by ( ) ( ) (7) v = + and the diection of v is always tangent to the path. j y i v = v v P = Fig. 3 Components of the velocity vecto x The acceleation vecto is compted as follows: dv d a = = + Sbstitting fo ( ) dt dt ( ) ( ) = = and = fom Eq.(3) gives 3/11

4 ME 01 DYNAMICS Chapte 1: Kinematics of a Paticle whee The tem ( ) ( ) ( ) ( ) ( + ) = a + a a = = + ( ) ( ) a = a = + is called the = d dt angla acceleation since it meases the change made in angla velocity ding an instant of time. Since a and a ae always pependicla as shown in Fig. 4, the magnitde of the acceleation is simply the positive vale of ( ) ( ) a = a + a ( ) ( ) a= + + The diection is detemined fom the vecto addition of the two components. In geneal, a will not be tangent to the path, Fig. 4. adial component tansvese component j y ( ) a = + i a ( ) a = Fig. 4 Components of the acceleation vecto P x Cylindical Coodinates Motion in thee-dimensions (Fig. 5) eqies a simple extension of the above fomlae to = + z Position: z v = + + z Velocity: z a = z Acceleation: ( ) ( ) z 4/11

5 ME 01 DYNAMICS Chapte 1: Kinematics of a Paticle Fig. 5 Cylindical coodinates in thee dimensional motion 5/11

6 ME 01 DYNAMICS Chapte 1: Kinematics of a Paticle EXAMPLE 1.17 (Textbook ) 6/11

7 ME 01 DYNAMICS Chapte 1: Kinematics of a Paticle 7/11

8 ME 01 DYNAMICS Chapte 1: Kinematics of a Paticle EXAMPLE 1.18 (Textbook ) 8/11

9 ME 01 DYNAMICS Chapte 1: Kinematics of a Paticle EXAMPLE 1.19 (Textbook ) 9/11

10 ME 01 DYNAMICS Chapte 1: Kinematics of a Paticle 10/11

11 ME 01 DYNAMICS Chapte 1: Kinematics of a Paticle 11/11