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 = ac /g with g = 9.8 m/s2

Rotational Motion Equations

Linear Linear Rotational Rotational

Symbol Unit Symbol Unit

Displacement d or x m Θ radians (rad)

Velocity v m/s ω rad/s

Acceleration a m/s2 α rad/s2

Mass m kg m kg

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 EquationsDescription Linear Angular

Displacement Δx = vΔt ΔΘ = ωΔt

Velocity v = Δx / Δt ω = ΔΘ / Δt

Acceleration a = Δv / Δt α = Δω / Δt

Final Velocity vf = vi + aΔt ωf = ωi + αΔt

Displacement Δx = vi Δt + ½ a (Δt)2 ΔΘ = ωi Δt + ½ α (Δt)2

Displacement Δx = ½ (vi + vf) Δt ΔΘ = ½ (ωi + ωf) Δt

(final velocity)2 vf2 = vi

2 + 2a (Δx) ωf2 = ωi

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)

S r

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)

Negative

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.• 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.

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

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

Description Equation

Definition of radian ΔΘ = Δs / r

Tangential velocity vt = r ω and vt = d / t

Tangential acceleration at = r α

Centripetal acceleration ac = r ω and ac = vt2 / r

Centripetal force Fc = mrω2 and Fc = mvt2 / r

* Remember that s = arc length