ENGINEERING & 
 
 MATHEMATICAL 
 
 SCIENCES LIBRARY 
 
 Class-mop notes on 
 ;lanar Kinematics 
 
 Irving Stringl 
 
 1 WEEK LOAN 
 
 iliforma 
 
 ;ional 
 
 ility
 
 
 UNIVERSITY OF CALIFORNIA 
 AT LOS ANGELES 
 
 Dr. and M rs . A .B. Pier 

 
 UNIVERSITY OF CALIFORNIA 
 
 1 
 
 CLASS-ROOM NOTES ON UNIPLANAR KINEMATICS
 
 QA 
 " ' 
 
 Library 
 
 UNIVERSITY OF CALIFORNIA. 
 
 CLASS-ROOM NOTES ON UNIPLANAR KINEMATICS. 
 
 Velocity in a Plane. A point moving in a plane is at any 
 instant determined in position by a pair of co-ordinates x, y or 
 r, ft, and its path may usually be defined by an equation, 
 y = f(x), or r = <p (ft). If p be the vector (directed magnitude) 
 from the origin to the point x, y or r, ft, this path is more 
 completely defined by the equations 
 
 p.= x -f- iy, y =/(x), 
 
 involving a complex function of the real variable x, or also, in 
 terms of tensor anff amplitude, by the equations 
 
 $v p = r cis ft, r -= <p (ft). 
 
 ^ The derivative of this vector p with respect to an equicres- 
 
 X cent variable t, which may be taken to represent time, is its rate, 
 
 ^ and is defined as the velocity of the moving point whose position 
 
 t^ at any instant it determines. In terms of x and y this velocity is 
 
 **^ dp dx . dy 
 
 Vdi ~~ di + Z HI 
 dx 
 and has a component in the direction of the ^r-axis, a 
 
 component ^ in the direction of the jy-axis, and the speed of 
 the point in its path is the tensor of this velocity, or 
 
 dp 
 dt 
 
 dx 
 
 4G2G20
 
 UNIPLANAR KINEMATICS. 
 
 On the other hand, in terms of r and tt, the velocity is 
 
 dp dr . u dH 
 
 and its components are 
 
 - cis ft = velocity in the direction of the vector p 
 dt 
 
 ja 
 ir cis ^ -j- -= velocity in a direction perpendicular to p, 
 
 and the speed of the point in its path is 
 
 dp 
 dt 
 
 djr\ 
 
 dt\ 
 
 {'ft 
 
 The components of velocity in two directions perpendicular 
 to each other as here determined are independent of each other 
 in the sense that neither involves any motion whatever in the 
 direction of the other. Obviously this is the case whatever be 
 the mode of resolution into two rectangular directions. 
 
 Velocity is compounded in still another important way by 
 introducing a second variable s, assumed to represent the length 
 of the path of the moving point measured from a fixed point. 
 In this form it is 
 
 dp ds dp 
 
 dl ~~ & ds 
 
 = v P', 
 where v = - -j- and /o' = : and since ds* = dx* + dy 1 , 
 
 dt 
 
 ds 
 
 = 
 
 If}'. 
 
 That | p f | = - i appears at once from the fact that 
 
 dp dx , .dy 
 r i 
 ds ds ds 
 
 and 
 
 dp 
 
 ds 
 
 Ids 
 
 \ds \
 
 UNIPLANAR KINEMATICS. 
 
 Hence v is the speed of the point along its path, as was asserted 
 
 dp 
 
 in the equation giving 
 
 above, and // is the velocity- 
 
 dt 
 
 direction, as is otherwise evident from the fact that 
 
 As=o ( As ) 
 
 has the direction of a tangent to the path of the point at the 
 instant considered. 
 
 Acceleration. The rate of change of velocity is called 
 acceleration. Expressed as a derivative it is 
 
 d 2 p d (ds dp} 
 
 dt' 1 dt \ dt ds j 
 
 d' 2 s dp d 2 p ( ds 1 * 
 
 T . 2 7 7 2 \ ~Jt f ' 
 
 at as as | a t j 
 
 .- ds dp d' l p 
 
 or lf ^ == v > ~T-=P> -jT = P > 
 
 dt ds ds 1 
 
 d 2 p dv 
 it is j- = p -\- if p , 
 
 and, like the analogous expressions for velocity, has two independ- 
 ent components .- p' and rfp", the former in the direction of the 
 tangent and in magnitude equal to the rate of change of speed, 
 the latter, as will be shown in the next paragraph, in a direction 
 perpendicular to the tangent and equal in magnitude to the 
 square of the speed multiplied by the curvature of the path at 
 the point considered. The first is called tangenital, the second 
 normal acceleration. 
 
 Radius of Curvature. L,et the equations of a plane curve be 
 
 ' 
 and let differentiation with respect to the arc s, estimated from a 
 
 dp d' 2 x 
 
 fixed point, be denoted by accents, - = p , -^- - = xr, etc. 
 
 as as 
 
 Then 
 
 462G20
 
 4 UNIPI^ANAR KINEMATICS. 
 
 p' = x 1 + z/ 
 
 p" = x" = i y" , 
 
 and since | /o | i 
 
 .-. x?* + y 2 == i 
 
 and *V'+yy'=o. 
 
 *v + yy - 
 
 p" ~ x m +y m 
 
 x'y"} 
 
 x"* + y? ' 
 
 which shows that p" is perpendicular to p', for z used as a 
 multiplier turns any line in the plane through a right angle. 
 And since | p' \ = i and | p" \ ' l = x' n + y"\ 
 .-. | p" | == | x "y> x'y" I . 
 
 But if q) be the angle between the Jt-axis and the tangent at the 
 extremity of p 
 
 limit Ax . limit Ay 
 
 T- = x= cos q>. and A = y= sin o>. 
 
 J^=o z/^ J^=o J^ * 
 
 or these expressions for sin (p and cos ^ may be deduced from 
 the equation (dxjds)*-}- (dylds)' l = i by assuming dxjds = cos ^, 
 inferring therefrom i (dx/ds)* = (dylds^j 1 = sin 2 q) and identi- 
 fying cp as the angle named. Then, by a second differentiation, 
 
 ,, dqt dq) 
 
 x '= sin (p . y = cos o> ~r~ , 
 ds ds 
 
 and thence, by cross multiplication with the previous equations, 
 
 x'y" x"y'= (cos 2 ^ + sin 2 <p) d ^= d ^. 
 
 ds as 
 
 dtp 
 
 is called the curvature of the curve at the point whose vector . 
 
 is p. Thus the assertion concerning the magnitude of the second, 
 or normal component of acceleration, at the close of the last 
 article, is verified. 
 
 The radius of curvature is the reciprocal of the curvature, or 
 
 R = ds - 
 
 dy- p"
 
 UNIPLANAR KINEMATICS. 5 
 
 Problems. Verify the following formulae for the deter- 
 mination of velocity and acceleration. 
 
 (i). If a point move with speed v in the curve y =. f(x}, 
 represented in Cartesian co-ordinates, prove that its velocity is 
 
 dp v_[i_ *'/'(>-)] 
 dt ~~ I/ i + [/''OrT]' 
 
 (2). If a point move with velocity v in the curve r =f(ff), 
 represented in polar co-ordinates, prove that its velocity is 
 
 (3). If v = speed, and R == radius of curvature, prove that 
 the tensor of acceleration is 
 
 dv 
 
 dt ) 
 
 In the following curves let s length of curve, = vectorial 
 angle, x = abscissa in a Cartesian system. If in each a point 
 move with speed v, determine the expressions for velocity, 
 acceleration, and radius of curvature. 
 
 (4.) p = j/V -(- $* -f- ia sinh -. 
 
 dp _ v (s -f- z) fl? 2 /o _ v 2 (a 1 ias} 
 
 (5) P = a e cis ^, the equiangular spiral. 
 
 dp v (i -(- m) 
 
 -.- - cis P. 
 
 "^ I/ i + w z 
 
 (6). p^ cis B, the spiral of Archimedes. 
 
 (?) Pz~ c i s ^' the reciprocal spiral. 
 
 (8). p = x -+- z'a^*", the logarithmic curve. 
 
 x 
 (9). p = x -\- zVcosh , the catenary. 
 
 IRVING STRINGHAM. 
 
 February,
 
 405 Hilgard Avenue, Los Angeles, CA 90024-1388 
 
 Return this material to tlje library 
 
 from which it was borrowed. 
 
 <jl APR19 
 
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