Physics 01, Lecture 17 Today s Topics q Rotation of Rigid Object About A Fixed Axis (Chap. 10.1-10.4) n Motion of Extend Object n Rotational Kinematics: n Angular Velocity n Angular Acceleration q Kinetic Energy of a Rotating Object q Moment of Inertia (More on Thursday) Motion Of An Extended Object q Extended Object = An object with size and shape A collection of point like objects (particles) a particle has no size/shape an extended object has mass, shape, described by m and position and size. Described by m, CM, moments of inertia (this week). q Motion of Rigid Object Rigid Object: Relative positions of all composing particles are fixed The shape of the rigid object does not change q Again and again, hope you h previewed! Ø Motion of Rigid Object = Motion of its Center of Mass + Rotation about the Center of Mass Translational Motion And Rotational Motion q Translational motion: The orientation of the object is unchanged during the motion. Motion of Rigid Object: Translation + Rotation q Rotational Motion: The object moves about an axis or center in circular fashion. = + 1
Review: Circular Motion q Circular motion: Angular velocity: = dθ/ Linear velocity: v = r, --- v always perpendicular to r q The acceleration has both a tangential and a centripetal components: The tangential component: a t = v/ = r/ = rα The centripetal component: a c = r = v /r Total acceleration: a = a C + a t q Rotation about a center: A group of particle together in circular motion a t v a c Rotation of Rigid Object About A Fixed Axis q Rotation about fixed axis is the simplest case of rotation Motion is described by change of quantity Angle θ v s θ = r sign convention +: counter clockwise - : clockwise (radian) à When rotating about a fixed axis, all elements on the rigid object are in circular motion with same angular speed: =dθ/ Quiz: Angular Velocity q Consider two points on a rigid object that rotates around a fixed axis as shown. Ø Which one has larger angular velocity? The Red dot, Blue dot, same. All points h the same angular velocity () Ø Which one has larger linear velocity? The Red dot, Blue dot, same. v = r Angular Velocity And Angular Acceleration q Angular Velocity () describes how fast an object rotstes, it has two components: Angular speed: and θ lim θ 0 = dθ direction of : + counter clockwise - clockwise è Angular velocity is a vector! (define direction next page) Ø All particles of the rigid object h the same angular velocity q Angular Acceleration (α): è Angular acceleration α α α and lim = d is also a vector! 0 Note: the similarity between (θ,,α) and (x, v, a)
Direction of Angular Velocity (Right-hand Rule) q The direction of angular velocity is define by a right-hand rule Practice: Right Hand Rule q What is the direction of angular velocity of this rotation? Quiz/Practice: Right Hand Rule Similarity Between (θ,,α) and (x,v,a) q The object rotates about the z axis as shown. Use right hand rule, what is the direction of its angular momentum? Towards left Towards right Up Down Into page Out of page α Rotation Angular Velocity: θ dθ Angular Acceleration: α d v a 1-D motion Velocity: x Acceleration v v dx a dv 3
Similarity Between (θ,,α) and (x,v,a): Kinematic Relationship 1-D motion Linear Velocity and Acceleration with Rotation For rigid object rotating about fixed axis The linear velocity has only tangential component, i.e. v = v t = r The linear acceleration can h both tangential and centripetal components: a t = dv/ = rd/ = rα a c = v /r = r, Rotational Kinetic Energy q General Kinetic Energy: KE i = ½ m i v i total kinetic energy: KE = Σ ½ m i v i q For an object rotating about a fixed axis: v i = r i KE = Σ ½ m i v i = ½ Σ m i (r i i ) = ½ (Σ m i r i ) i = ½ I Moment of Inertia q Moment of Inertia of an object about an axis Moment of Inertia: I m i r i another form: I r dm whole object (unit of I : kgm ) axis Rotational Kinetic Energy : KE rot = 1 I Ø I depends on rotation axis, total mass, and mass distribution. 4
Quick Quiz The picture below shows two different dumbbell shaped objects. Object A has two balls of mass m separated by a distance L, and object B has two balls of mass m separated by a distance L. Which of the objects has larger moment of inertia for rotations around x-axis? A. A. B. B. C. They h the same moment of inertia Case A : ml = ml Case B: m L = ml Exercise: Moment of Inertia of a Uniform Ring q Image the hoop is divided into a number of small segments, m 1 q These segments are equidistant from the axis I = m i r i = ( m i ) R = MR Or calculus form: I = r dm = R dm = MR axis Exercise: Moment of Inertia of a Uniform Disc Ø Area density: α= M/A = M/(πR ) Ø Mass element at r: dm=αda= α rdθdr (review some basic geometric calculus of you are in question) q Now Moment of inertia I = disc R π r dm = r α r dr dθ 0 0 R = πα r 3 dr = πα R4 = 1 MR 0 R axis r da Moments Of Inertial Of Various Objects I = m i r i (= r dm) 5
Quick Quiz Compare a thick wall and a thin wall cylinder of same mass and outer radius. Which one has moment of inertia around the axis shown? Quick Quiz Order the following objects, all having the same R and M, according their moments of inertia around there respective axis as shown. ( 1=largest) R 4 1 Thin wall cylinder Thick wall cylinder 3 3 1 6