PHY126 Summer Session I, 2008 Most of information is available at: http://nngroup.physics.sunysb.edu/~chiaki/PHY126-08 including the syllabus and lecture slides. Read syllabus and watch for important announcements. Homework assignment for each chapter due nominally a week later. But at least for the first two homework assignments, you will have more time. All the assignments will be done through MasteringPhysics, so you need to purchase the permit to use it. Some numerical values in some problems will be randomized. In addition to homework problems and quizzes, there is a reading requirement of each chapter, which is very important.

Chapter 10: Rotational Motion Movement of points in a rigid body All points in a rigid body move in circles about the axis of rotation Axis of rotation z orbit of point P A rigid body has a perfectly definite and unchanging shape and size. Relative position of points in the body do not change relative to one another. y P In this specific example on the left, the axis of rotation is the z-axis. rigid body x

Movement of points in a rigid body (cont’d) At any given time, the 2-d projection of any point in the object is described by two coordinates ( r , q ) In our example, 2-d projection onto the x-y plane is the right one. y length of the arc from the x-axis s: P s = rq where s,r in m, and q in rad(ian) r q A complete circle: s = 2pr 360o = 2p rad x 57.3o= 1 rad 1 rev/s = 2p rad/s 1 rev/min = 1 rpm

Angular displacement, velocity and acceleration At any given time, the 2-d projection of any point in the object is described by two coordinates ( r , q ) y Angular displacement: Dq = q2 - q1 in a time interval Dt = t2 – t1 P at t2 Average angular velocity: r rad/s P at t1 q2 Instantaneous angular velocity: q1 x rad/s w < (>) 0 (counter) clockwise rotation

Angular displacement, velocity and acceleration (cont’d) At any given time, the 2-d projection of any point in the object is described by two coordinates ( r , q ) y Average angular acceleration: rad/s2 P at t2 Instantaneous angular acceleration: r rad/s2 P at t1 q2 q1 x

Correspondence between linear & angular quantities Displacement Velocity Acceleration

Case for constant acceleration (2-d) Consider an object rotating with constant angular acceleration a0 Eq.(1) Eq.(2)

Case for constant acceleration (2-d) (cont’d) Eliminating t from Eqs.(1) & (2): Eq.(1) Eq.(1’) Eqs.(1’)&(2)

Vectors Vectors in 3-dimension : unit vector in x,y,z direction Consider a vector: x Also Inner (dot) product

Small change of radial vector Rotation by a small rotation angle Note:

Relation between angular & linear variables y unit tangential vector unit vector in y direction unit radial vector r is const. A point with Fixed radius x unit vector in x direction

Relation between angular & linear variables (cont’d) y unit vector in y direction A point with Fixed radius x unit vector in x direction tangential component radial component

Relation between angular & linear variables (cont’d) Example How are the angular speeds of the two bicycle sprockets in Fig. related to the number of teeth on each sprocket? The chain does not slip or stretch, so it moves at the same tangential speed v on both sprockets: The angular speed is inversely proportional to the radius. Let N1 and N2 be the numbers of teeth. The condition that tooth spacing is the same on both sprockets leads to: Combining the above two equations:

Description of general rotation Why it is not always right to define a rotation by a vector original rotation about x axis rotation about y axis y y y x x x rotation about x axis original rotation about y axis z z z y y y x x x The result depends on the order of operations

Description of general rotation Up to this point all the rotations have been about the z-axis or in x-y plane. In this case the rotations are about a unit vector where is normal to the x-y plane. But in general, rotations are about a general direction. Define a rotation about by Dq as: In general but if infinitesimally small, we can define a vector by RH rule If the axes of rotation are the same,

Kinetic energy & rotational inertia A point in a Rigid body x rotational inertia/ moment of inertia axis of rotation

Kinetic energy & rotational inertia (cont’d) More precise definition of I : A point in a Rigid body density volume element x Compare with: And remember conservation of energy: rotation axis

Kinetic energy & rotational inertia (cont’d) Moment of inertia of a thin ring (mass M, radius R) (I) y linear mass density ds x rotation axis (z-axis)

Kinetic energy & rotational inertia (cont’d) Moment of inertia of a thin ring (mass M, radius R) (II) rotation axis (y-axis) y ds x

Kinetic energy & rotational inertia (cont’d) Table of moment of inertia

Kinetic energy & rotational inertia (cont’d) Tables of moment of inertia

Parallel axis theorem The axis of rotation is parallel to the z-axis y rotation axis through P x, y measured w.r.t. COM y dm r P y-b x-a d COM b O a x x total mass rotation axis through COM COM: center of mass

Parallel axis theorem (cont’d) y Knowing the moment of inertia about an axis through COM (center of mass) of a body, the rotational inertia for rotation about any parallel axis is : rotation axis through P y dm r P y-b x-a d COM b x O a x rotation axis through COM

Correspondence between linear & angular quantities displacement velocity acceleration mass kinetic energy

Exercises Problem 1 A meter stick with a mass of 0.160 kg is pivoted about one end so that it can rotate without friction about a horizontal axis. The meter stick is held in a horizontal position and released. As it swings through the vertical, calculate the change in gravitational potential energy that has occurred; the angular speed of the stick; the linear speed of the end of the stick opposite the axis. Compare the answer in (c) to the speed of a particle that has fallen 1.00m, starting from rest. y cm Solution (a) 1.00 m cm 0.500 m x

Problem 1 (cont’d) (b) (c) (d)

Problem 2 The pulley in the figure has radius R and a moment of inertia I. The rope does not slip over the pulley, and the pulley spins on a frictionless axle. The coefficient of kinetic friction between block A and the tabletop is The system is released from rest, and block B descends. Block A has mass mA and block B has mass mB. Use energy methods to calculate the speed of Block B as a function of distance d that it has descended. Solution A I Energy conservation: y B x d

Problem 3 You hang a thin hoop with radius R over a nail at the rim of the hoop. You displace it to the side through an angle b from its equilibrium position and let it go. What is its angular speed when it returns to its equilibrium position? y Solution pivot point R R x the origin = the original location of the center of the hoop