Physics 207: Lecture 17, Pg 1 Lecture 17 (Catch up) Goals: Chapter 12 Chapter 12  Introduce and analyze torque  Understand the equilibrium dynamics of an extended object in response to forces  Employ “conservation of angular momentum” concept Assignment: l HW8 due March 17 th l Thursday, Exam Review

Physics 207: Lecture 17, Pg 2 Rotational Motion: Statics and Dynamics l Forces are still necessary but the outcome depends on the location from the axis of rotation l This is in contrast to the translational motion and acceleration of the center of mass. Here the position of these forces doesn’t matter (doesn’t alter the physics we see) l However: For rotational statics & dynamics: we must reference the specific position of the force relative to an axis of rotation. It may be necessary to consider more than one rotation axis! l Vectors remain the key tool for visualizing Newton’s Laws

Physics 207: Lecture 17, Pg 3 Angular motion can be described by vectors l With rotation the distribution of mass matters. Actual result depends on the distance from the axis of rotation. l Hence, only the axis of rotation remains fixed in reference to rotation. We find that angular motions may be quantified by defining a vector along the axis of rotation. l We can employ the right hand rule to find the vector direction

Physics 207: Lecture 17, Pg 4 The Angular Velocity Vector The magnitude of the angular velocity vector is ω. The angular velocity vector points along the axis of rotation in the direction given by the right-hand rule as illustrated above. As  increased the vector lengthens

Physics 207: Lecture 17, Pg 5 So: What makes it spin (causes  ) ?  NET = |r| |F Tang | ≡ |r| |F| sin  l If a force points at the axis of rotation the wheel won’t turn l Thus, only the tangential component of the force matters l With torque the position & angle of the force matters F Tangential a r F radial F  A force applied at a distance from the rotation axis gives a torque F radial F Tangential  r =|F Tang | sin 

Physics 207: Lecture 17, Pg 6 Rotational Dynamics: What makes it spin?  NET = |r| |F Tang | ≡ |r| |F| sin  l Torque is the rotational equivalent of force Torque has units of kg m 2 /s 2 = (kg m/s 2 ) m = N m F Tangentiala r F radial F  A force applied at a distance from the rotation axis   NET = r F Tang = r m a Tang = r m r  = (m r 2 )  For every little part of the wheel

Physics 207: Lecture 17, Pg 7 For a point mass  NET = m r 2  l This is the rotational version of F NET = ma Moment of inertia, is the rotational equivalent of mass. Moment of inertia, I ≡  i  m i r i 2, is the rotational equivalent of mass. If I is big, more torque is required to achieve a given angular acceleration. F Tangentiala r F randial F  The further a mass is away from this axis the greater the inertia (resistance) to rotation (as we saw on Thursday)  NET = I 

Physics 207: Lecture 17, Pg 8 Rotational Dynamics: What makes it spin?  NET = |r| |F Tang | ≡ |r| |F| sin  l A constant torque gives constant angular acceleration if and only if the mass distribution and the axis of rotation remain constant. F Tangential a r F radial F  A force applied at a distance from the rotation axis gives a torque

Physics 207: Lecture 17, Pg 9 Torque, like , is a vector quantity Magnitude is given by (1)|r| |F| sin   (2)|F tangential | |r| (3)|F| |r perpendicular to line of action | l Direction is parallel to the axis of rotation with respect to the “right hand rule” And for a rigid object  = I  r F F cos(90 °  ) = F Tang. r F F radial F a r F r  90 °  r sin  line of action

Physics 207: Lecture 17, Pg 10 Exercise Torque Magnitude A. Case 1 B. Case 2 C. Same l 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  or the tangential force component times perpendicular distance L L F F axis case 1case 2

Physics 207: Lecture 17, Pg 11 Example: Rotating Rod Again l A uniform rod of length L=0.5 m and mass m=1 kg is free to rotate on a frictionless pin passing through one end as in the Figure. The rod is released from rest in the horizontal position. What is the initial angular acceleration  ? L m

Physics 207: Lecture 17, Pg 12 Example: Rotating Rod l A uniform rod of length L=0.5 m and mass m=1 kg is free … What is its initial angular acceleration ? 1. For forces you need to locate the Center of Mass CM is at L/2 ( halfway ) and put in the Force on a FBD 2. The hinge changes everything! L m mg  F = 0 occurs only at the hinge but  z = I  z = - r F sin 90° at the center of mass and I End  z = - (L/2) mg and solve for  z

Physics 207: Lecture 17, Pg 13 Work (in rotational motion) Consider the work done by a force F acting on an object constrained to move around a fixed axis. For an infinitesimal angular displacement d  :where dr =R d   dW = F Tangential dr  dW = (F Tangential R) d   dW =  d  and with a constant torque) We can integrate this to find: W =  (  f  i  l Analog of W = F  r W will be negative if  and  have opposite sign ! axis of rotation R F dr =Rd  dd 

Physics 207: Lecture 17, Pg 14 Statics Equilibrium is established when In 3D this implies SIX expressions (x, y & z)

Physics 207: Lecture 17, Pg 15 Example l Two children (60 kg and 30 kg) sit on a horizontal teeter-totter. The larger child is 1.0 m from the pivot point while the smaller child is trying to figure out where to sit so that the teeter-totter remains motionless. The teeter-totter is a uniform bar of 30 kg its moment of inertia about the support point is 30 kg m 2. l Assuming you can treat both children as point like particles, what is the initial angular acceleration of the teeter-totter when the large child lifts up their legs off the ground (the smaller child can’t reach)? l For the static case:

Physics 207: Lecture 17, Pg 16 Example: Soln. l Draw a Free Body diagram (assume g = 10 m/s 2 ) l 0 = 300 d x N x 0 – 600 x 1.0 0= 2d + 1 – 4 d = 1.5 m from pivot point 30 kg 1 m 60 kg 30 kg 0.5 m 300 N 600 N N

Physics 207: Lecture 17, Pg 17 Classic Example: Will the ladder slip? Not if

Physics 207: Lecture 17, Pg 18 Classic Example: Will the ladder slip?

Physics 207: Lecture 17, Pg 19 EXAMPLE Will the ladder slip?

Physics 207: Lecture 17, Pg 20 EXAMPLE Will the ladder slip?

Physics 207: Lecture 17, Pg 21 Angular Momentum: l We have shown that for a system of particles, momentum is conserved if l What is the rotational equivalent of this (rotational “mass” times rotational velocity)? angular momentum is conserved if

Physics 207: Lecture 17, Pg 22 Angular momentum of a rigid body about a fixed axis: l 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: l Even if no connecting rod we can define an L z i r1r1 r3r3 r2r2 m2m2 m3m3 j m1m1  vv2vv2 vv1vv1 vv3vv3 Using v i =  r i, we get ( r i and v i, are perpendicular)

Physics 207: Lecture 17, Pg 23 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. 00 z FF z

Physics 207: Lecture 17, Pg 24 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. 00 z FF z No External Torque so L z is constant L i = L f  I  i i = I  f  ½ mR 2  0 = ½ 2mR 2  f

Physics 207: Lecture 17, Pg 25 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 ? Top view: before after d v M I I FF

Physics 207: Lecture 17, Pg 26 Example: Throwing ball from stool What is the angular speed  F of the student-stool system after they throw the ball ? l Process: (1) Define system (2) Identify Conditions (1) System: student, stool and ball (No Ext. torque, L is constant) (2) Momentum is conserved L init = 0 = L final = -m v d + I  f Top view: before after d v M I I FF

Physics 207: Lecture 17, Pg 27 Angular Momentum as a Fundamental Quantity l The concept of angular momentum is also valid on a submicroscopic scale l Angular momentum has been used in the development of modern theories of atomic, molecular and nuclear physics l 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

Physics 207: Lecture 17, Pg 28 Lecture 17 Assignment: l HW7 due March 17th l Thursday: Review session