Lecture 17 Goals Relate and use angle, angular velocity & angular acceleration Identify vectors associated with angular motion Introduce Rotational Inertia Introduce Rotational Kinetic Energy Recognize superposition of rotational and translational motions 1

Chapter 10: Rotational Motion Rotation is common in the world around us. Many ideas developed for translational motion are transferable.

Analogous to the linear case Rotational Variables Rotation about a fixed axis: Consider a disk rotating about an axis through its center: Recall :   Analogous to the linear case

Rotational Variables Rotation about a fixed axis: If a wheel is turning faster or slower then w is changing with time Thus “angular acceleration” is  

But on any linear path with constant acceleration Rotational Variables... At a point a distance R away from the axis of rotation, the tangential motion: s (arc) =  R But on any linear path with constant acceleration   R v = w R s  On any curvilinear path with constant tangential acceleration On a circular path with constant acceleration

Rotational Variables...   R v = w R s 

Rotational Variables... vT (tangential) =  R aT =  R  At a point a distance R away from the axis of rotation, the tangential motion: s (arc) =  R vT (tangential) =  R aT =  R   R v = w R s 

Overview (with comparison to 1-D kinematics) Angular Linear Rotational motion has parallels with linear motion

Angular velocity is a vector quantity The axis of rotation is unique! The direction of the axis defines the direction of the angular velocity vector. Example If rotation axis parallel to the x axis, then Use the right hand rule to determine sign CCW is positive CW is negative

Exercise Rotational Definitions A friend at a party sees a disk spinning and says “Ooh, look! There’s a wheel with a negative w and positive a!” Which of the following is a true statement about the wheel? ? The wheel is spinning counter-clockwise and slowing down. The wheel is spinning counter-clockwise and speeding up. The wheel is spinning clockwise and slowing down. The wheel is spinning clockwise and speeding up.

System of Particles (Distributed Mass): Until now, we have considered the behavior of modest systems (with a relatively small number of masses). But real objects have distributed mass ! For example, consider a simple rotating disk and two equal mass m plugs at distances r and 2r from the rotation axis. How do the velocities compare? How do the angular velocities compare? How do the kinetic energies compare? w 1 2

System of Particles (Distributed Mass): 1 K= ½ m v2 = ½ m (w r)2 w 2 K= ½ m (2v)2 = ½ m (w 2r)2 Twice the radius, four times the kinetic energy Rigid object: Mass and radius vary but w is always the same

Rotation & Kinetic Energy Consider the simple rotating system shown below. (Assume the masses are attached to the rotation axis by massless rigid rods). The kinetic energy of this system will be the sum of the kinetic energy of each piece: K = ½ m1v12 + ½ m2v22 + ½ m3v32 + ½ m4v42 m4 m1 r4  r1 m3 r2 r3 m2

Rotation & Kinetic Energy Notice that v1 = w r1 , v2 = w r2 , v3 = w r3 , v4 = w r4 So we can rewrite the summation: We define a new quantity, moment of inertia or I, and write: r1 r2 r3 r4 m4 m1 m2 m3 

Calculating Moment of Inertia where r is the distance from the mass to the axis of rotation. Example: Calculate the moment of inertia of four point masses (m) on the corners of a square whose sides have length L, about a perpendicular axis through the center of the square: m m L m m

Calculating Moment of Inertia... For a single object, I depends on the rotation axis! Example: I1 = 4 m R2 = 4 m (21/2 L / 2)2 I1 = 2mL2 I2 = mL2 I = 2mL2 m m L m m

Calculating Moment of Inertia... For a discrete collection of point masses we found: For a continuous solid object we have to add up the mr2 contribution for every infinitesimal mass element dm. An integral is required to find I : dm r

Moments of Inertia... Some examples of I for solid objects: Solid sphere of mass M and radius R, about an axis through its center. I = 2/5 M R2 R Thin spherical shell of mass M and radius R, about an axis through its center. Look it up… R

Connection with motion... So for a solid object rotating about its center of mass and whose CM is moving: VCM 

For Thursday All of Chapter 10 HW8 due Monday