Rotational Kinematics

The need for a new set of variables

0We have talked about things in linear motion and in purely rotational movement, but many object both spin and move linearly0 Rolling balls0 Planets in orbit0 Tennis balls or baseballs or volleyballs after they have

been hit0Most 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

0 360o=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

Quantity Linear Rotational Connection

Position

Displacement

Acceleration

1st kinematic

2nd kinematic

3rd kinematic

Centripetal acceleration

x (or y) Ɵ

Δx ΔƟ Δx=rΔƟ

Angular Velocity

0Variable is ω (omega)0Linear velocity is the change in position (Δx) over the

change in time 0Angular velocity is ω=Δ /Ɵ Δt0 If you divide each side of the equation x = r by Ɵ Δt,

you get v = rω0Similarly if you manipulate the equation for tangential

velocity you get ω=2π/T

Angular Acceleration

0Variable is α (alpha)0Linear acceleration is the change in velocity (Δv) over

the change in time 0Angular acceleration is α=Δ /ω Δt0 If you divide each side of the equation v = rω by Δt,

you get a = rα

Flip back to the table

Quantity Linear Rotational Connection

Position

Displacement

Velocity

Acceleration

1st kinematic

2nd kinematic

3rd kinematic

x (or y) Ɵ

Δx ΔƟ Δx=rΔƟ

v=Δx/Δt ω=Δ /Ɵ Δt v=rω

a=Δv/Δt α=Δω/Δt a=rα

A demo…..

0https://prettygoodphysics.wikispaces.com/Rotational+Motion%2C+Torque%2C+Angular+Momentum

Rotational Kinematics Equations

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

2

2 2

1

2

2

f o

o

f o

t

t t

Centripetal Acceleration

2ca r

Flip back to the table

Quantity Linear Rotational Connection

Position

Displacement

Velocity

Acceleration

1st kinematic

2nd kinematic

3rd kinematic

Centripetal acceleration

x (or y) Ɵ

Δx ΔƟ Δx=rΔƟ

v=Δx/Δt ω=Δ /Ɵ Δt=2π/T v=rω

a=Δv/Δt α=Δω/Δt a=rα

0fv v at 0f t

21

2ox v t at 21

2ot t

2 20 2fv v a x

2 20 2f

2

c

va

r

2ca r

Examples0A knight swings a mace of radius 1m in two

complete revolutions. What is the translational displacement of the mace?

Examples0A 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?

Examples0A 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/s2 to a maximum speed of 15,000 rpms. What is the maximum speed of the saw in radians per second?

Examples0How long does it take the saw to

reach its maximum speed?

Examples0How many complete rotations does

the saw make while accelerating to its maximum speed?

Examples0A 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?

N m 2r

fs N

Fy 0

f s mg0

N mg

fs

mg

N

fs N

fs mg

s g

2r

9.8m / s2

(4 r / s)2 3.5m0.18

sm 2rmg

Fx mac

m 2r

ac 2r

ac v 2

r

v r