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PHY126 Summer Session I, 2008

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.

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PHY126 Summer Session I, 2008

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  1. 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.

  2. Chapter 10: Rotational Motion • All points in a rigid body move in circles about the axis of rotation Axis of rotation z 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. orbit of point P Movement of points in a rigid body y P In this specific example on the left, the axis of rotation is the z-axis. rigid body x

  3. 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) Movement of points in a rigid body (cont’d) 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

  4. 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: Angular displacement, velocity and acceleration r rad/s P at t1 q2 Instantaneous angular velocity: q1 x rad/s w < (>) 0 (counter) clockwise rotation

  5. 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: Angular displacement, velocity and acceleration (cont’d) r rad/s2 P at t1 q2 q1 x

  6. Correspondence between linear & angular quantities

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

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

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

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

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

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

  13. v 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: Relation between angular & linear variables (cont’d) 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:

  14. Why it is not always right to define a rotation by a vector rotation about x axis rotation about y axis original y y y x Description of general rotation 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

  15. 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 Description of general rotation but if infinitesimally small, we can define a vector by RH rule • If the axes of rotation are the same,

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

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

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

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

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

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

  22. 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 y-b P x-a d COM Parallel axis theorem b a O x x total mass rotation axis through COM COM: center of mass

  23. 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 y-b P x-a d COM Parallel axis theorem (cont’d) b x a O x rotation axis through COM

  24. Correspondence between linear & angular quantities

  25. 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. Exercises y cm Solution (a) 1.00 m cm 0.500 m x

  26. Problem 1 (cont’d) (b) (c) (d)

  27. 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

  28. 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

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