Question

Note: Final answer from the textbook is at the bottom. I need help understanding the process on getting to that answer

EXERCISE 4.28 A cylinder of radius r rolls without slipping inside a semicylindrical cavity. Point P is collinear with the vertical centerline when the vertical angle e locating the cylinder's center C is zero. Derive expressions for the velocity and acceleration of point P in terms of o, e, and ē. Exercise 4.28

Answer 1

Solm. H is convenient to orient the Body fixed Reference frame such that at the Instant of Interest, the a xraxis is Tangent to the path of the center and the y-ax & Intersects the center of curvature, Movement of wheel cyinder to the sigut then cossosfonds to Jo vê, ão vãt I : -w.kar.com - r-R ; u Because there is no sliping at the contact point A it also must be that, V - W xócA. we maten the two expression for Je to obtain a relation for w, and then differcutiate it with the result that W = VR . 7 W - Now that we have characterized wą and a we may evaluate the velocity and aceleration of point' si Jp = Be two tople au ? + (-X) { x ce-sing)? - ecos f5] -v[ 1-ecoso Ji - ulei sind). I -O ap-cac+28ple - wapple aul + V2 + ( v ) x (esing)? - (e. cos f)]] - () { ce.sings I - (ecos f)]]

ap = 1 [l-e.com A) – rece simm] P + [ v using + to expreses rea in terms of vertical angle '0', we must establish the relation Between d8 O. for this we return to the Norseip Condition, it was stated that point op. is on the vestical centerline when 0=0. The coual-arelength form of the worselip constraint requires the arclength Measured along the chunder and along the cavity from the celorent points of contact to the tri tial points be the game, which leads to ro= R(22-0) = $-01- Put this value of & Inean D. we get v = (--) .o [(itę.com®]87 - ing)"3J.) Proved. and aceperation is sāp = (R-o) ë (1+4.6053) - %1% -1).02. sing] I + (R-8) [-0 % sind +221 + e - eR). Los o] ] Proved,