2Important Terms and Concepts Radians/degrees: how to convert from one to anotherRotational motion: what is it?Arc lengthAngular displacement (in radians)Angular velocity (ω: omega)Angular acceleration (α: alpha)Rotational motion equations vs. linear motion equationsTangential velocity/accelerationCentripetal accelerationCauses of rotational motion (gravity/torque)Conservation of Angular Momentum
3Basic 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 π360 Θd
10Sample 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?
11Sample 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?
14A mysterious planet 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?
15A 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.
16Conservation of Angular Momentum The velocity is the TANGENTIAL VELOCITYSo, the unit for angular momentum L is:“In any closed system, the total amount of angular momentum is conserved.”
25Figure SkatersI 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.
26Figure SkatersInstead 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.
27Figure SkatersSo, 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.