Rotational Kinematics

The need for a new set of variables 0 We have talked about things in linear motion and in purely rotational movement, but many object both spin and move linearly 0 Rolling balls 0 Planets in orbit 0 Tennis balls or baseballs or volleyballs after they have been hit 0 Most rotational kinematics variables will be Greek letters

Radians 0 So far we have talked about everything in degrees, but it now makes sense to switch to radians because a radian relates an angle (rotation) to a distance on the circle (linear) 0 A radian is defined as the measure of a central angle  that makes an arc length s equal in length to the radius r of the circle. 0 If we call the arc length (the linear movement) x, then x = rƟ o =2π radian so 1 revolution = 2π and T would be the time it takes to go 2π 0 We will fill in the table at the end of the notes as we go so flip there now.

Table QuantityLinearRotationalConnection Position Displacement Acceleration 1 st kinematic 2 nd kinematic 3 rd kinematic Centripetal acceleration x (or y)Ɵ ΔxΔx ΔƟΔƟ Δx=rΔƟ

Angular Velocity 0 Variable is ω (omega) 0 Linear velocity is the change in position (Δx) over the change in time 0 Angular velocity is ω=ΔƟ/Δt 0 If you divide each side of the equation x = rƟ by Δt, you get v = rω 0 Similarly if you manipulate the equation for tangential velocity you get ω=2π/T

Angular Acceleration 0 Variable is α (alpha) 0 Linear acceleration is the change in velocity (Δv) over the change in time 0 Angular acceleration is α=Δω/Δt 0 If you divide each side of the equation v = rω by Δt, you get a = rα

Flip back to the table QuantityLinearRotationalConnection Position Displacement Velocity Acceleration 1 st kinematic 2 nd kinematic 3 rd kinematic x (or y)Ɵ ΔxΔx ΔƟΔƟ Δx=rΔƟ v=Δx/Δtω=ΔƟ/Δt v=rω a=Δv/Δtα=Δω/Δta=rα

A demo… l+Motion%2C+Torque%2C+Angular+Momentum

Rotational Kinematics Equations 0 Using these equations and relationships we can write the rotational kinematics equations

Centripetal Acceleration

Flip back to the table QuantityLinearRotationalConnection Position Displacement Velocity Acceleration 1 st kinematic 2 nd kinematic 3 rd kinematic Centripetal acceleration x (or y)Ɵ ΔxΔx ΔƟΔƟ Δx=rΔƟ v=Δx/Δtω=ΔƟ/Δt=2π/T v=rω a=Δv/Δtα=Δω/Δta=rα

Examples 0 A knight swings a mace of radius 1m in two complete revolutions. What is the translational displacement of the mace?

Examples 0 A compact disc player is designed to vary the disc’s rotational velocity so that the point being read by the laser moves at a linear velocity of 1.25 m/s. What is the CD’s rotational velocity in rev/s when the laser is reading information on an inner portion of the disc at a radius of 0.03 m?

Examples 0 A carpenter cuts a piece of wood with a high powered circular saw. The saw blade accelerates from rest with an angular acceleration of 14 rad/s 2 to a maximum speed of 15,000 rpms. What is the maximum speed of the saw in radians per second?

Examples 0 How long does it take the saw to reach its maximum speed?

Examples 0 How many complete rotations does the saw make while accelerating to its maximum speed?

Examples 0 A safety mechanism will bring the saw blade to rest in 0.3 seconds should the carpenter’s hand come off the saw controls. What angular acceleration does this require? How many complete revolutions will the saw blade make in this time?

The Rotor +y +x An amusement park ride called the Rotor spins with an angular speed of 4 radians/s. It has a radius of 3.5 m. What is the minimum coefficient of friction so the riders don’t slip? fsfs mg N