Angular Acceleration

angular displacement

• the angle that the object rotates through during some time interval

• ∆θ = θƒ-θi

centripetal or radical acceleration

• makes the object spin around in a circle without affecting its speed 

• ac = v² / r

ac = v² / r 

what is the equation for centripetal or radical acceleration (ac)? 

average angular velocity in a time interval

• the angular displacement divided by the time interval

• ω. = ∆θ / ∆t

angular acceleration (α)

• when a rotating objects angular velocity is not constant

  • object spinning slowly @ first and then gets faster and faster over time

• α = ∆ω / ∆t = ωƒ-ωi / tƒ - ti

α = ∆ω / ∆t = ωƒ-ωi / tƒ - ti

what is the equation for angular acceleration (α)? 

uniform circular motion

• the only acceleration is the centripetal acceleration

  • spinning @ constant speed (angular velocity ω is constant), angular accel = 0

    • object has no angular accel but it does have a centripetal acceleration that points towards the center of the circle

      • centripetal accel is what makes the object go round and round

Non-unifrom circular motion

• there is also a tangential component to the acceleration (that makes the acceleration speed up/increase) 

• spinning faster and faster

  • object has centripetal accel that keeps it moving in circle, as object speeds up the larger the speed, the larger the centripetal accel

tangential acceleration

• if object is speeding up or slowing down, in addition to its angular accel it has a ___?

• aT = dv / dt = d(rω) / dt = r × (dω / dt) = rα

• aT = rα

aT = rα

what is the equation for tangential acceleration (aT)?

rotational kinematics

• when a rigid object rotates with constant angular acceleration, we describe its motion using 4 kinematics equations:

  • ∆θ = ωi∆t + ½ α∆t²

  • ωf = ωi + α∆t

  • ωf² = ωi² + 2α∆θ

  • ∆θ =. ½ (ωi + ωf) ∆t

• same equations as the kinematics formula for 1D linear motion but replace:

  • x (position) → θ (angular position)

  • v (velocity) → ω (angular velocity) 

  • a (accel) → α (angular accel)