Two-Dimensional Rotational Kinematics 8.01 W09D1

Today’s Reading Assignment: W09D1 Young and Freedman: 1.10 (Vector Products) 9.1-9.6, 10.5

Rigid Bodies Springs or human bodies are non-rigid bodies. A rigid body is an extended object in which the distance between any two points in the object is constant in time. Springs or human bodies are non-rigid bodies.

Demo Center of Mass and Rotational Motion of Baton

Overview: Rotation and Translation of Rigid Body Demonstration: Motion of a thrown baton Translational motion: external force of gravity acts on center of mass Rotational Motion: object rotates about center of mass

Recall: Translational Motion of the Center of Mass Total momentum of system of particles External force and acceleration of center of mass 6

Main Idea: Rotational Motion about Center of Mass Torque produces angular acceleration about center of mass is the moment of inertial about the center of mass is the angular acceleration about center of mass

Two-Dimensional Rotational Motion Fixed Axis Rotation: Disc is rotating about axis passing through the center of the disc and is perpendicular to the plane of the disc. Plane of Motion is Fixed: For straight line motion, bicycle wheel rotates about fixed direction and center of mass is translating Hyphen added in title 8

Cylindrical Coordinate System Unit vectors

Rotational Kinematics for Point-Like Particle

Rotational Kinematics for Fixed Axis Rotation A point like particle undergoing circular motion at a non-constant speed has an angular velocity vector (2) an angular acceleration vector

Fixed Axis Rotation: Angular Velocity Angle variable SI unit: Angular velocity Vector: Component magnitude direction 12

Concept Question: Angular Speed Object A sits at the outer edge (rim) of a merry-go-round, and object B sits halfway between the rim and the axis of rotation. The merry-go-round makes a complete revolution once every thirty seconds. The magnitude of the angular velocity of Object B is half the magnitude of the angular velocity of Object A . the same as the magnitude of the angular velocity of Object A . twice the the magnitude of the angular velocity of Object A . impossible to determine.

Example: Angular Velocity Consider point-like object rotating with velocity tangent to the circle of radius r as shown in the figure below with The angular velocity vector points in the direction, given by

Fixed Axis Rotation: Angular Acceleration SI unit Vector: Component: Magnitude: Direction: Address font size issue on subscript in figure 15

Rotational Kinematics: Integral Relations The angular quantities are exactly analogous to the quantities for one-dimensional motion, and obey the same type of integral relations Example: Constant angular acceleration Second slide for constant example 16

Concept Question: Rotational Kinematics The figure shows a graph of ωz and αz versus time for a particular rotating body. During which time intervals is the rotation slowing down? 0 < t < 2 s 2 s < t < 4 s 4 s < t < 6 s None of the intervals. Two of the intervals. Three of the intervals.

Table Problem: Rotational Kinematics A turntable is a uniform disc of mass m and a radius R. The turntable is initially spinning clockwise when looked down on from above at a constant frequency f . The motor is turned off and the turntable slows to a stop in t seconds with constant angular deceleration. a) What is the direction and magnitude of the initial angular velocity of the turntable? b) What is the direction and magnitude of the angular acceleration of the turntable? c) What is the total angle in radians that the turntable spins while slowing down?

Summary: Circular Motion for Point-like Particle Use plane polar coordinates: circle of radius r Unit vectors are functions of time because direction changes Position Velocity Acceleration

Rigid Body Kinematics for Fixed Axis Rotation Kinetic Energy and Moment of Inertia

Rigid Body Kinematics for Fixed Axis Rotation Body rotates with angular velocity and angular acceleration

Divide Body into Small Elements Body rotates with angular velocity, angular acceleration Individual elements of mass Radius of orbit Tangential velocity Tangential acceleration Radial Acceleration For accuracy and agreement with the notes, make the last item the magnitude of the radial acceleration. 22

Rotational Kinetic Energy and Moment of Inertia Rotational kinetic energy about axis passing through S Moment of Inertia about S : SI Unit: Continuous body: Rotational Kinetic Energy: Resized parentheses on unit; in general, having the superscripts extend above the delimiter looks bad. 23

Discussion: Moment of Inertia How does moment of inertia compare to the total mass and the center of mass? Different measures of the distribution of the mass. Total mass: scalar Center of Mass: vector (three components) Moment of Inertia about axis passing through S: (nine possible moments) Why nine? There’s no way I can find nine unless the integrand allows “dm x y” or something similar, which is part of an outer product, which is a tensor, and I know we agreed to stay away from these beasts. 24

Concept Question All of the objects below have the same mass. Which of the objects has the largest moment of inertia about the axis shown? (1) Hollow Cylinder (2) Solid Cylinder (3)Thin-walled Hollow Cylinder 25

Concept Question

Concept Question

Worked Example: Moment of Inertia for Uniform Disc Consider a thin uniform disc of radius R and mass m. What is the moment of inertia about an axis that pass perpendicular through the center of the disc?

Strategy: Calculating Moment of Inertia Step 1: Identify the axis of rotation Step 2: Choose a coordinate system Step 3: Identify the infinitesimal mass element dm. Step 4: Identify the radius, , of the circular orbit of the infinitesimal mass element dm. Step 5: Set up the limits for the integral over the body in terms of the physical dimensions of the rigid body. Step 6: Explicitly calculate the integrals. “dm” in text changed to italics. I’m glad to see the use of Times Roman instead of sans-serif. 29

Worked Example: Moment of Inertia of a Disc Consider a thin uniform disc of radius R and mass m. What is the moment of inertia about an axis that pass perpendicular through the center of the disc? We lose students here 30

Parallel Axis Theorem Rigid body of mass m. Moment of inertia about axis through center of mass of the body. Moment of inertia about parallel axis through point S in body. dS,cm perpendicular distance between two parallel axes. 31

Table Problem: Moment of Inertia of a Rod Consider a thin uniform rod of length L and mass M. Odd Tables : Calculate the moment of inertia about an axis that passes perpendicular through the center of mass of the rod. Even Tables: Calculate the moment of inertia about an axis that passes perpendicular through the end of the rod. Shouldn’t it be “passes perpendicular to the rod through its center of mass”? Do this at board

Summary: Moment of Inertia Moment of Inertia about S: Examples: Let S be the center of mass rod of length l and mass m disc of radius R and mass m Parallel Axis theorem:

Table Problem: Kinetic Energy of Disk A disk with mass M and radius R is spinning with angular speed ω about an axis that passes through the rim of the disk perpendicular to its plane. The moment of inertia about the cm is (1/2)M R2. What is the kinetic energy of the disk?

Concept Question: Kinetic Energy A disk with mass M and radius R is spinning with angular speed ω about an axis that passes through the rim of the disk perpendicular to its plane. Moment of inertia about cm is (1/2)M R2. Its total kinetic energy is: (1/4)M R2 ω2 (1/2)M R2 ω2 3. (3/4)M R2 ω2 4. (1/4)M Rω2 5. (1/2)M Rω2 6. (1/4)M Rω

Summary: Fixed Axis Rotation Kinematics Angle variable Angular velocity Angular acceleration Mass element Radius of orbit Moment of inertia Parallel Axis Theorem

Simple Pendulum Simple Pendulum: bob of mass m hanging from end of massless string of length I pivoted at S. Angular velocity Angular acceleration Kinetic energy of rotation

Worked Example: Simple Pendulum A simple pendulum is released from rest at an angle . Find the angular speed at angle

Worked Example: Simple Pendulum Energy Diagram The energy diagrams for the pendulum are shown in the figure below

Worked Example: Simple Pendulum Velocity Kinetic energy Initial energy Final energy Conservation of energy

Table Problem: Pulley System and Energy Using energy techniques, calculate the speed of block 2 as a function of distance that it moves down the inclined plane using energy techniques. Let IP denote the moment of inertia of the pulley about its center of mass. Assume there are no energy losses due to friction and that the rope does slip around the pulley.

Next Reading Assignment: W09D2 Young and Freedman: 1.10 (Vector Product) 10.1-10.2, 10.5-10.6 ; 11.1-11.3