Wednesday: Review session Physics 207, Lecture 16, Oct. 27 Goals: Chapter 12 Extend the particle model to rigid-bodies Understand the equilibrium of an extended object. Understand rotation about a fixed axis. Employ “conservation of angular momentum” concept Assignment: HW7 due Oct. 29 Wednesday: Review session 1

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

Rotational Dynamics: What makes it spin? A force applied at a distance from the rotation axis gives a torque a FTangential F NET = |r| |FTang| ≡ |r| |F| sin f Fradial r Only the tangential component of the force matters. With torque the position of the force matters Torque is the rotational equivalent of force Torque has units of kg m2/s2 = (kg m/s2) m = N m A constant torque gives constant angular acceleration iff the mass distribution and the axis of rotation remain constant.

Torque is a vector quantity Magnitude is given by |r| |F| sin q or, equivalently, by the |Ftangential | |r| or by |F| |rperpendicular to line of action | Direction is parallel to the axis of rotation with respect to the “right hand rule” And for a rigid object  = I a r F F cos(90°-q) = FTang. Fradial a q 90°-q r sin q line of action

Exercise Torque Magnitude In which of the cases shown below is the torque provided by the applied force about the rotation axis biggest? In both cases the magnitude and direction of the applied force is the same. Remember torque requires F, r and sin q or the tangential force component times perpendicular distance L F F Case 1 Case 2 Same L axis case 1 case 2

Rotational Dynamics: What makes it spin? A force applied at a distance from the rotation axis FTangential a r Fradial F NET = |r| |FTang| ≡ |r| |F| sin f Torque is the rotational equivalent of force Torque has units of kg m2/s2 = (kg m/s2) m = N m NET = r FTang = r m aTang = r m r a = (m r2) a For every little part of the wheel

For a point mass NET = m r2 a and inertia FTangential a r Frandial F The further a mass is away from this axis the greater the inertia (resistance) to rotation This is the rotational version of FNET = ma Moment of inertia, I ≡ m r2 , (here I is just a point on the wheel) is the rotational equivalent of mass. If I is big, more torque is required to achieve a given angular acceleration.

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

Moments of Inertia An integral is required to find I : dm r 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 : r dm Some examples of I for solid objects: R L r dr Solid disk or cylinder of mass M and radius R, about perpendicular axis through its center. I = ½ M R2 Use the table…

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 recognize the quantity, moment of inertia or I, and write: r1 r2 r3 r4 m4 m1 m2 m3 

Exercise Rotational Kinetic Energy We have two balls of the same mass. Ball 1 is attached to a 0.1 m long rope. It spins around at 2 revolutions per second. Ball 2 is on a 0.2 m long rope. It spins around at 2 revolutions per second. What is the ratio of the kinetic energy of Ball 2 to that of Ball 1 ? ¼ ½ 1 2 4 Ball 1 Ball 2

Work & Kinetic Energy: Recall the Work Kinetic-Energy Theorem: K = WNET This applies to both rotational as well as linear motion. So for an object that rotates about a fixed axis For an object which is rotating and translating

Angular Momentum: We have shown that for a system of particles, momentum is conserved if What is the rotational equivalent of this? angular momentum

Angular momentum of a rigid body about a fixed axis: Consider a rigid distribution of point particles rotating in the x-y plane around the z axis, as shown below. The total angular momentum around the origin Is the sum of the angular momentum of each particle: Even if no connecting rod we can deduce an Lz ( ri and vi , are perpendicular) i r1 r3 r2 m2 m3 j m1  v2 v1 v3 Using vi =  ri , we get

Example: Two Disks A disk of mass M and radius R rotates around the z axis with angular velocity 0. A second identical disk, initially not rotating, is dropped on top of the first. There is friction between the disks, and eventually they rotate together with angular velocity F. z F z 0

Example: Two Disks A disk of mass M and radius R rotates around the z axis with initial angular velocity 0. A second identical disk, at rest, is dropped on top of the first. There is friction between the disks, and eventually they rotate together with angular velocity F. No External Torque so Lz is constant Li = Lf  I wi i = I wf  ½ mR2 w0 = ½ 2mR2 wf 0 z F

Example: Throwing ball from stool A student sits on a stool, initially at rest, but which is free to rotate. The moment of inertia of the student plus the stool is I. They throw a heavy ball of mass M with speed v such that its velocity vector moves a distance d from the axis of rotation. What is the angular speed F of the student-stool system after they throw the ball ? M v F d I I Top view: before after

Example: Throwing ball from stool What is the angular speed F of the student-stool system after they throw the ball ? Process: (1) Define system (2) Identify Conditions (1) System: student, stool and ball (No Ext. torque, L is constant) (2) Momentum is conserved Linit = 0 = Lfinal = -m v d + I wf M v F d I I Top view: before after

Angular Momentum as a Fundamental Quantity The concept of angular momentum is also valid on a submicroscopic scale Angular momentum has been used in the development of modern theories of atomic, molecular and nuclear physics In these systems, the angular momentum has been found to be a fundamental quantity Fundamental here means that it is an intrinsic property of these objects

Fundamental Angular Momentum Angular momentum has discrete values These discrete values are multiples of a fundamental unit of angular momentum The fundamental unit of angular momentum is h-bar Where h is called Planck’s constant

Intrinsic Angular Momentum photon

Angular Momentum of a Molecule Consider the molecule as a rigid rotor, with the two atoms separated by a fixed distance The rotation occurs about the center of mass in the plane of the page with a speed of

Angular Momentum of a Molecule (It heats the water in a microwave over) E = h2/(8p2I) [ J (J+1) ] J = 0, 1, 2, ….

Physics 207, Lecture 16, Oct. 27 Assignment: HW7 due Oct. 29 Wednesday: Review session 1