1. Rotational Motion Equations All linear equations can be applied to rotational motion. We use Greek letters to designate rotational motion. To calculate G-forces, use Gforce = a c /g with g = 9.8 m/s 2

2. Rotational Motion Equations Linear Rotational SymbolUnitSymbolUnit Displacementd or xmΘradians (rad) Velocityvm/sωrad/s Accelerationam/s 2 αrad/s 2 Massmkgm Symbols and terminology for both linear and rotational motion:

3. Rotational Motion Equations The rotational motion symbols for angular velocity, acceleration, and displacement can be substituted directly into the linear kinematic equations.

4. Rotational Motion Equations DescriptionLinearAngular DisplacementΔx = vΔtΔΘ = ωΔt Velocityv = Δx / Δtω = ΔΘ / Δt Accelerationa = Δv / Δtα = Δω / Δt Final Velocityv f = v i + aΔtω f = ω i + αΔt DisplacementΔx = v i Δt + ½ a (Δt) 2 ΔΘ = ω i Δt + ½ α (Δt) 2 DisplacementΔx = ½ (v i + v f ) ΔtΔΘ = ½ (ω i + ω f ) Δt (final velocity) 2 v f 2 = v i 2 + 2a (Δx)ω f 2 = ω i 2 + 2α (ΔΘ)

5. Rotational Motion Equations What about Angles… A new measurement of angles (Radians) is introduced to describe objects that are rotating. A Radian is defined as: The ratio of “arc traversed” divided by the radius  =  s/r It is a pure number – no units Denoted by symbol  (theta)

6. Rotational Motion Equations More about angles… Positive angles represent counter-clockwise rotation Negative angles represent clockwise rotation Hint: Visualize your car moving forward – the tires will rotate counter clockwise Zero is the positive “x” axis. 2  radians describes one full rotation (360 degrees) Positive

7. More about angles… So why invent radians? What’s wrong with degrees? Consider the following question: If my tire is making 21.2 revolutions per minute, how fast is my car moving? (12.25” radius) Solution: 21.2 RPM is 42.4  radians per minute. The distance is 42.4  * Radius / minute. 1631.7 inches/min ~ 1.54 MPH Solution without radians: Calculate circumference of tire Times 21 Calculate.2 arc length Add together to get distance Now have distance / minute Net is that radians makes calculations easier.

8. Rotational Motion Equations Practice Problem… A girl sitting on a merry-go-round moving counterclockwise through an arc length of 2.50 meters. If the angular displacement was 1.67 rad, how far is she from the center of the merry-go-round? Solution:  =  s/r 1.67 = 2.5 meters /r r = 1.497 meters

9. Rotational Motion Equations Angular Speed… Linear Speed – how fast is it moving? Velocity v =  distance /  time m/s Angular Speed – how fast is it rotating? Angular Speed =  /  time Symbol is:  “omega” Angular Speed  =  /  t radians/second

10. Rotational Motion Equations There are also equations only for rotational motion using the symbols for angular velocity, angular displacement, and angular acceleration.

11. Rotational Motion Equations DescriptionEquation Definition of radianΔΘ = Δs / r Tangential velocityv t = r ω and v t = d / t Tangential accelerationa t = r α Centripetal accelerationa c = r ω and a c = v t 2 / r Centripetal forceF c = mrω 2 and F c = mv t 2 / r * Remember that s = arc length