Rotational Motion WHS Lee Wignall

Important Terms and Concepts Radians/degrees: how to convert from one to another Rotational motion: what is it? Arc length Angular displacement (in radians) Angular velocity (ω: omega) Angular acceleration (α: alpha) Rotational motion equations vs. linear motion equations Tangential velocity/acceleration Centripetal acceleration Causes of rotational motion (gravity/torque) Conservation of Angular Momentum

Basic Quantities Converting from degrees to radians: So far we’ve done everything in degrees, but now we will make a change to radians for rotational motion. Why? Because things usually “turn” more than just one revolution (360 degrees). Converting from degrees to radians: 2π = θr so, θr = θd π 360 Θd 180

Arc Length Arc length (m) Radius (m) angle (radians)

ω = Δθ Δt Angular Velocity Linear equation for velocity: change in distance change in time Angular velocity: change in angle “omega”: radians/s “angular displacement” ω = Δθ Δt

ω = Δθ Δt Angular Velocity Angular velocity: change in angle change in time “angular displacement” “omega” ω = Δθ Δt COUNTER CLOCKWISE = positive direction (+) CLOCKWISE = negative direction (-)

α = Δω Δt Angular Acceleration Linear equation for acceleration: change in velocity change in time Angular acceleration: change in angular velocity “alpha”: radians/s2 α = Δω Δt

Angular vs. Linear Motion Equations

Sample Rotational Motion Problem #1: A wheel is slowly turning at 1.0 rad/s when an outside force causes the angular velocity to increase to 4.0 rad/s. If it takes 5 full revolutions of the wheel to reach it’s final angular velocity, what was the angular acceleration provided by the force?

Sample Rotational Motion Problem #2: A helicopter blade accelerates at 35.5 rad/s2. If the initial angular velocity of the blade was 5 rad/s, how long will it take the blades to complete 1500 turns? What will be its angular velocity at that point? If it takes an angular velocity of 2000 rad/s to achieve liftoff, how many seconds will that take?

Homework Angular Displacement: page 247 (1-4) Angular Speed: page 248 (1-4) Angular Acceleration: page 250 (1-3) Angular Kinematics: page 252 (1-5)

Tangential Velocity and Acceleration Radius (m) vt = rω Angular velocity (rad/s) Tangential velocity (m/s) at = rα Angular acceleration (rad/s2) Tangential acceleration (m/s2)

Homework Finish Today’s Work: Tangential speed page 255 (1-4) Angular Displacement: page 247 (1-4) Angular Speed: page 248 (1-4) Angular Acceleration: page 250 (1-3) Angular Kinematics: page 252 (1-5) Finish Today’s Work: Tangential speed page 255 (1-4) Tangential acceleration: page 256 (1-3)

Advanced Rotational Motion Problems

A mysterious planet with radius of 3 A mysterious planet with radius of 3.6 x 109 m has a gravitational pull of 15 m/s2 but it’s spinning so fast that a 50 kg. rock at the equator no longer falls downward. At what latitudes (north and south) will the rock have a downward acceleration of 10 m/s2?

A 200 gram ball on a 0. 75 meter string has an angular velocity of 1 A 200 gram ball on a 0.75 meter string has an angular velocity of 1.5 rad/s. The person spinning it supplies a force that provides an acceleration of 2.0 rad/s2. The string will break when the tension reaches 150 N. How many seconds until the string breaks? At what angle will the ball go flying off at? Assume you the ball begins at 0 degrees and spins counter-clockwise.

Conservation of Angular Momentum The velocity is the TANGENTIAL VELOCITY So, the unit for angular momentum L is: “In any closed system, the total amount of angular momentum is conserved.”

Formation of Galaxies Horsehead Nebula

Formation of Galaxies Crab Nebula

Formation of Galaxies

Formation of Galaxies The Earth and tossed pizza dough do the same thing.

Figure Skaters I is the “moment of inertia” which is also called the radial mass. It’s a measure of how much mass is how far away from the axis of rotation.

Figure Skaters Instead of thinking about moments of inertia and angular velocity, we can think about this spinning skater in terms of radius and tangential velocity. It’s easier.

Figure Skaters So, in order to conserve angular momentum L, if we DECREASE the radial distance for the overall mass, then we will experience an INCREASE in the tangential velocity, meaning we spin faster.