Lecture 10, oct. 08, 2013 Announcement Mid-term: Date: 22 oct 2013 (Tue) Time: 9:30 – 11:15 am venue: PHYSI 1 IOA: science center Ll Content: things learned before todays class Format: 5 long questions Exercise/tutorial session at MMW 71 5 (TA: ZHAO Saisai) will be merged with the session at SC Ll HW 4 is launched today and due next Thursday (Oct. 17, 2013) Reminder: HW 3 is due this Thursday Review 1 . The Law of Conservation of Linear Momentum: = pxf pyl = pyf Pzi = Pzf 2.
Two-body collision process: Linear momentum is conserved for a closed and isolated two-body system during collision because there is no net external force. Total kinetic energy of the system of two colliding bodies may or may not be conserved. If it is conserved, the collision is called an elastic collision. If the kinetic energy of the system is not conserved, the collision is called an inelastic collision. Completely inelastic collision: deformation is totally irreversible, so the two objects always will come to a common velocity in one dimension. 3.
Systems of varying mass: Linear momentum is conserved for the entire system of the main body and exhausted mass, if net external force is zero; Use Impulse-Linear momentum theorem when there is et external force; Two key quantities: mass exhaust rate (R = -dM/dt) and relative velocity between the exhausted mass and the main body (vrel = vrocket – vexhaust) v + dv Ch6. Rotation and Moment 6. 1 6. 2 6. 3 6. 4 6. 5 6. 6 6. 7 The Rotational Variables Kinetic Energy of Rotation Moment of Inertia Torque (Moment) Bending Moment Newton’s Second Law for Rotation Work and Rotational Kinetic Energy 6. Rolling as Translation and Rotation Combined 6 Y Klnetlc Energy 0T Rolling 6. 10 The Forces of Rolling 6. 11 Angular Momentum 6. 12 The Angular Momentum of a Rigid Body Rotating About a Fixed Axis 6. 3 Conservation of Angular Momentum 6. 1 The Rotational Variables We consider the rotation of a rigid body about a fixed axis. o A rigid body is a body that can rotate with all its parts locked together and without any change in its shape. o A fixed axis means that the rotation occurs about an axis that does not move.
In pure rotation, every point of the body executes a circular motion about the fixed axis. Credit: Sunil Kumar Singh Angular Position is the angle of the reference line relative to a fixed direction. o A reference line is fixed in the body, perpendicular to the rotation axis and rotating with the body. A fixed direction represents the zero angular position, is perpendicular to the rotation axis and does not rotate with the body. For convenience, the reference direction like x-axis of the coordinate system serves to represent fixed direction.
The angle between reference direction and rotating arm (OP) at any instant is the angular position of the particle (B). Geometrically, the angular position B is given by: where s is the length of a circular arc that extends Trom tne x-axls (tne zero angular position) to the reference line, and r is the radius of the circle. Note: Angular position defined here has a unit of radians rad): Angular Displacement is equal to the difference of angular positions at two instants of rotational motion. o An angular displacement in the counterclockwise direction is positive, and one in the clockwise direction is negative. Referring to the right diagram, the angular displacement AB is given by: The Average Angular Velocity wavg of the rotating body in the time interval At from tl to t2 is defined by: wavg = At The (instantaneous) Angular Velocity w is the limit of wavg as O: ABdB w = lim dt The Average Angular Acceleration aavg of the rotating body in the time interval At from tl tot2 is defined by: 2 -WI a avg = The (instantaneous) Angular Acceleration a is the limit of aavg as O: a = lim Analogy Equations of Motion for Constant Linear Acceleration and for Constant Angular Acceleration The angular quantities (velocity and acceleration) are also vector quantities like their linear counter parts and follow vector rules of addition and multiplication, with the notable exception of finite angular displacement. o Finite angular displacement does not follow the rule of vector addition strictly. o It can be shown that addition of angular displacement depends on the order in which they are added. Performing the rotations in one order gives one result, (3), while reversing the order gives a different result, (5. Credit: Benjamin Crowell We can represent an Angular Velocity as a vector pointing along the axis of rotation by applying “Right hand rule”: o Holding the axis of rotation with right hand in such a manner that the direction of the curl of fingers is along the direction of the rotation. o The direction of extended thumb (along y-axis in the fgure) represents the direction of angular velocity. o Dlsplacement Is equal to tne OITTerence of angular positions at two instants of rotational motion. The linear and angular velocity are related in the following vector form: v = wxr In the case of the uniform circular motion, the speed (v) of the particle is constant.
This implies that angular velocity is also constant: The relation between linear and angular acceleration can be found by direct differentiation: angular acceleration dv d dr In the case of the uniform circular motion, both linear speed (v) and angular speed (w) are constant and the linear acceleration is also constant: a = w 2r Pointing toward the origin along the radial direction – centripetal acceleration 6 2 Klnetlc Energy 0T Rotatlon Consider a rigid body rotates with constant angular velocity. Its Rotational Kinetic Energy K is the total kinetic energy of the particles in the body given by: K=zrnt Vi2 10 20 2 20 i Moment of Inertia or Rotational Inertia, I 6. 3 Moment of Inertia The Moment of Inertia I ofa rotating body is defined by: o The moment of inertia is a constant for a particular rigid body and a particular rotation axis. The rotational kinetic energy can be expressed as: 2 If a rigid body consists ofa few particles, we can calculate its moment of inertia about a given rotation axis with the formula: Find the product mr2 of each particle and then sum the roducts. If a rigid body consists of a great many adjacent particles (continuous mass distribution), the moment of inertia can be defined by: 1=fr2dm where dm is the infinitesimal mass element, depending on the actual mass distribution. The Moment of Inertia of a rotating body involves not only its mass but also how that mass is distributed. A long rod is much easier to rotate about (a) its central (longitudinal) axis than about (b) an axis through its centre and perpendicular to its length.
Parallel-Axis Theorem o The moment of inertia I ofa body of mass M about an axis is related to its moment of inertia bout a parallel axis through its centre of mass by: 1=1com + Mh 2 where h is the perpendicular distance between the given axis and the axis through the centre of mass. (see the proof in HRW) The moment of inertia formulas of some common geometric shapes 6. 3 Moment 0T Inertla Application of Parallel-Axis Theorem o One may apply the parallel-axis theorem to determine the moment of inertia I of rod about an axis perpendicular to the rod and passing through one of its ends as shown in the diagram. I = I com + Md 2 = ML+MC]O 12 020 ML2 3 where M is the mass of the rod Sample problem Reading Chapter 10, sections 1-7 of the textbook