Lecture 17 Goals: Chapter 12 Understand the equilibrium dynamics of an extended object in response to forces Analyze rolling motion Employ “conservation of angular momentum” concept Assignment: HW7 due March 25th After Spring Break Tuesday: Catch up 1

Rotational Dynamics: What makes it spin? A “special” net force is necessary and that action depends on the location from the axis of rotation For translational motion and acceleration the position of the force doesn’t matter (doesn’t change the physics we see) IMPORTANT: For rotational motion and angular acceleration: we always reference the specific position of the force relative to the axis of rotation. Vectors are simply a tool for visualizing Newton’s Laws

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

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.

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 A constant torque gives constant angular acceleration iff the mass distribution and the axis of rotation remain constant.

Angular motion can be described by vectors Recall that linear motion involves x, y and/or z vectors Angular motion can be quantified by defining a vector along the axis of rotation. The axis of rotation is the one set of points that is fixed with respect to a rotation. Use the right hand rule

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

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

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

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 f or the tangential force component times perpendicular distance L F axis case 1 case 2 (A) case 1 (B) case 2 (C) same

Example: Rotating Rod 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 (A) its angular speed when it reaches the lowest point ? (B) its initial angular acceleration ? (C) initial linear acceleration of its free end ? L m

Example : Rolling Motion A cylinder is about to roll down an inclined plane. What is its speed at the bottom of the plane ? M q h v ? Ball has radius R

Example : Rolling Motion A cylinder is about to roll down an inclined plane. What is its speed at the bottom of the plane ? Use Work-Energy theorem M q h v ? Ball has radius R Mgh = ½ Mv2 + ½ ICM w2 Mgh = ½ Mv2 + ½ (½ M R2 )(v/R)2 = ¾ Mv2 v = 2(gh/3)½

Motion Again consider a cylinder rolling at a constant speed. Both with |VTang| = |VCM | Rotation only VTang = wR Sliding only 2VCM VCM CM CM CM VCM

Angular Momentum: We have shown that for a system of particles, momentum is conserved if What is the rotational equivalent of this (rotational “mass” times rotational velocity)? 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

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

Lecture 17 Assignment: HW7 due March 25th Thursday: Review session 1