Rotational Motion Chapter 8: Rotational Motion In physics we distinguish two types of motion for objects: Translational Motion (change of location): Whole object moves through space. Rotational Motion - object turns around an axis (axle); axis does not move. (Wheels) Rotational Motion Study rotational motion of solid object with fixed axis of rotation (axle) Have angular versions of the quantities we studied in translational motion - position, distance, velocity, acceleration Later look at angular versions of force, mass, momentum, and kinetic energy. Use Greek letters for most angular quantities We will measure angular position in revolutions: Counterclockwis e (CCW): positive rotation Clockwise (CW): negative rotation Angular Position Linear Distance d vs. Angular distance Δθ Ranking: Rolling Cups For a point at radius R on the wheel, d = 2πRΔθ for Δθ in revolutions R Which of the cups will roll in the straightest path? Which of the cups will roll in the most curved path? 1
Angular Velocity: ω Avg. Angular Velocity = # Revolutions/ (Time Taken) ω = Δθ / t Unit: Revolutions/s or Revolutions/min (RPM) Sign convention : ω is positive for counterclockwise rotation, negative for clockwise rotation Tangential Velocity Every spot on a rotating object has both angular velocity and tangential velocity. v t = Δd/Δt = 2πRΔθ/Δt = 2πRω RΔθ R Speed in Circular Motion Rotational Speed ω: Rev.s per second Tangential speed v t : distance per second Two objects can have the same rotational speed, but different tangential speeds! Two wheels are connected by a chain that doesn t slip. Which wheel has the higher rotational speed? Example: Gears Which wheel has the higher tangential speed for a point on its rim? Angular Acceleration Simple vs. Complex Objects Change in angular velocity -> angular acceleration! However, even if angular velocity is constant, each point also has centripetal acceleration (due to change in direction of v t ) Model motion with just Position Model motion with position and Rotation 2
Rotational Inertia Rotational inertia depends on Total mass of the object Distribution of the mass relative to axis Farther the mass is from the axis of rotation, the larger the rotational inertia. Rotational inertia ~ (mass) x (axis_distance)2 Rotational Inertia Depends upon the axis around which it rotates Easier to rotate pencil around an axis passing through it. Harder to rotate it around vertical axis passing through center. Hardest to rotate it around vertical axis passing through the end. Example: Hoop vs. Disk Imagine rolling a hoop and a disk of equal mass down a ramp. Which one would win? Which one is easier to rotate (i.e., has less rotational inertia)? Torque Torque is the rotational analog of force. Depends on: Magnitude of Force Direction of force Lever arm torque = lever arm x force Units of N m Examples of Lever Arm Example: Pedaling a Bicycle Lever arm is amount of perpendicular distance to where the force acts. 3
Revisiting Newton s Laws 1: Need a linear force to change an object s linear motion Need a torque to change an object s rotational motion Equilibrium: Linear: ΣF = 0 Rotational: Στ = 0 2: Translational acceleration ~ force, and ~ 1/mass Angular acceleration ~ torque, and ~ 1/rotational inertia Example: See-Saw Balancing 4 m? m Ranking Which meter stick requires the most torque to hold up the weight? Center of Mass Average position of all the mass in an object is called the center of mass (CM) of object. Average position of the weight distribution is called the center of gravity (CG). When gravity is constant (usually the case), these two locations are the same. Stability: Balancing Acts Objects are stable, as long as their CG is above the base of the object. Example Three trucks are parked on a slope. Which truck(s) tip over? 4
Centripetal Force Centripetal means towards the center. Whenever an object moves along a circular path, there must be a force on that object in the direction of the center of the circle. In such a case, the force is said to be centripetal Centripetal Force Any force directed toward a fixed center is called a centripetal force. Centripetal means center-seeking or toward the center. F = mv 2 /r r = radius of circle v = tangential velocity Example: The spin cycle! Example You are riding at the very edge of a merrygo-round with a radius of 2 m. Your friend runs alongside, pushing the merry-go-round so that it s tangential speed is 3 m/s. a. What force is keeping you from sliding off? b. If you have a mass of 75 kg, what is the strength of that force? Centrifugal Force (by XKCD) 5
Angular Momentum Recall: Linear momentum = mass x velocity Angular momentum = rotational inertia x rotational velocity L = I ω Need an impulse to change linear momentum need a torque to change angular momentum! Angular Momentum Special case: for an object that is small relative to its axis of rotation (planet in its orbit, bug on a turntable) Example: Merry-go-round What is the angular momentum of our 75 kg person going 3 m/s on the merry-goround with radius of 2 m? Angular momentum L = mvr Units: kg m 2 /s Conservation of Angular Momentum If no external net torque acts on a rotating system, then the angular momentum of that system remains constant. Examples Conservation of angular momentum plays a big role in astronomy, because it relates tangential speed (or orbital speed) to radius (or orbital distance). Formation of stars, planetary systems, and galaxies Moon s orbit around the Earth 6