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Rolling Lecturer: Tran Thi Ngoc Dung dungttn@gmail.com HCMUT What does it mean “Without sliding”? The sphere is not sliding relative to the bar if the velocities of the two adjacent points A and B are the same where A is on the sphere and B is on the bar Without sliding : v A / ground v B / ground or vA vB The velocity of A relative to B is : v A / B v A / ground v B / ground A B Ground If there is ' sliding': v A v B and v A / B Rotation about a moving axis = Rolling A cylinder is rolling on the ground CM v cm A Ground Two types: - Rolling with sliding - Rolling without sliding B CM A B Rolling Rolling Rolling v cm = Translation of a point of the object = Translation of the center of mass = Translation of point A + Rotation about the axis through this point + Rotation about the axis through CM + Rotation about the axis through A Velocity in rolling motion Velocity of a point M of the rolling object v M / ground v M / cm v cm / ground Velocity of a point M in the rotation about the axis through CM v M / cm CM v E / cm E v A / cm Ground CM v cm A B D v B / cm | v A / cm | R | v D / cm | R R | v E / cm | v M / ground v M / cm v cm / ground y x CM R v A / cm Ground v cm Av CM Rolling without sliding, D v CM v D / gr v D / cm B Projecting (1)on the x axis : vA because the ground is not moving, vB=0 v A v A / cm v cm R v cm R v cm : angular velocity in rotational motion : angular acceleration Vcm,acm: speed and acceleration of the CM R a cm (1) How to Solve Problems ‘Rolling without Sliding’ Method Rolling = Translation of the center of mass ma cm Fnet ,ext + Rotation about the axis through CM Icm net,ext / cm Step Write Eq of Motion of the CM Step Write Eq of Rotation about the axis through CM Step Find relation between acm and How to Solve Problems ‘Rolling without Sliding’ Method Rolling = Translation of point A having vA=0 + Rotation about the axis through A I A net,ext / A Step Write Eq of Rotation about the axis through A => Step Find relation between acm and =>acm y x N CM f A B W Method Example A cylinder of mass m, radius R rolls without sliding on an inclined plane Find a) acceleration of CM b) angular acceleration c) friction force d) condition for having ‘rolling without sliding’ Step1 Eq of motion of CM : ma cm mg N f friction Pr ojecting on x and y axis (1) / x : ma cm mg sin f friction (2) (1) / y : mg cos N (3) N mg cos (3' ) Step2 Eq of Rotation about the axis though CM : Icm f friction R (4) y Step3 Re lation between a cm and : a cm R (1) (5) I cm From (4), (5) : a cm f friction (6) R I mg sin (3) (6) : (m cm2 )a cm mg sin a cm Icm R m R from(2) : f friction mg sin ma cm x N CM f A B W mg sin Icm m R a cm / R a cm x f f friction mg sin ma cm a) solid cylinder : Icm b) solid sphere : Icm a cm mR 2 f friction mg sin mR f friction mg sin f friction A B W a cm g sin ; b) Hoop : Icm mR g sin ; N CM N mg cos a cm g sin ; y mg sin • condition for having ' rolling without sliding ' f fric f s max f fric s N Cylinder mg sin s mg cos tan 3s y x Method Example A cylinder of mass m, radius R rolls without sliding on an inclined plane Find a) acceleration of CM b) angular acceleration c) friction force d) condition for having ‘rolling without sliding’ N CM f A B W Eq of Rotation about the axis though CM : I A mgR sin mgR sin IA Parallel Axis Theory : I A Icm mR mR 2 g sin 3R a cm R g sin g sin 3R a cm g sin Find f fric.from a ) ma cm mg sin f friction or b)Icm f friction R Rolling without and with sliding For rolling without sliding : f friction f static f s max v cm R a cm R Work of static friction force is zero For rolling with sliding : f friction k N v cm R a cm R Wfric k N.s Kinetic Energy in Rolling K mvcm 2 K.E of Translational motion of CM Icm2 2 K E of Rotational Motion about the axis throughCM Homework 1) A cylinder rolls down a plane inclined at = 50° What is the minimum value of the coefficient of static friction for which the cylinder will roll without slipping? (Answer 0.40) 2) For a hoop rolling down an incline, (a) what is the force of friction, (b) what is the maximum value of tan for which the hoop will roll without slipping? (Answers (a) f mgsin, (b) tan = 2µs ) ... ' sliding' : v A v B and v A / B Rotation about a moving axis = Rolling A cylinder is rolling on the ground CM v cm A Ground Two types: - Rolling with sliding - Rolling without sliding. .. friction R Rolling without and with sliding For rolling without sliding : f friction f static f s max v cm R a cm R Work of static friction force is zero For rolling with sliding : f... v cm A Ground Two types: - Rolling with sliding - Rolling without sliding B CM A B Rolling Rolling Rolling v cm = Translation of a point of the object = Translation of the center of mass