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

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:

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

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)

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)

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

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 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.

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 = meters

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

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

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