An Overview of Rotational Motion C 8 via

An Overview of Rotational Motion (C 8) via a Comparison to Translational Motion (C 2 -4, 6 -7) Note these essentially interchangeable terms: linear & translational angular & rotational Rotational Motion Translational Motion Kinematics (C 2) Kinematics (8. 1 -3) n Basic variables Δx: displacement (m) Δθ: angular disp (radians) v: velocity (m/s) ω: angular velocity (rad/s) : angular accel (rad/s 2) a: acceleration (m/s 2) n Constant eq’ns: n Constant a eq’ns: ω f = ω i + Δt v f = v i + a Δt Δθ = ωi Δt + ½ Δt 2 Δ x = v i Δ t + ½ a Δ t 2 Δθ = ωf Δt - ½ Δt 2 Δ x = v f Δ t - ½ a Δ t 2 ωf 2 = ωi 2 + 2 Δθ vf 2 = vi 2 + 2 a Δx Δx = (vi + vf) Δt Δθ = (ωi + ωf) Δt 2 2

Connection Between Translational and Rotational Variables Define radian: angle subtended by an arc of the unit circle, where l = r, so θ = l /r = 1 radian • no actual units on θ – radians & degrees are not true units, but labels to tell which scale was used • but can convert between them by 1 rev = 360° = 2π radians = 6. 28 radians so 1 radian ≈ 57° To convert from a linear to a rotational quantity: • since 1 rev = 2π (in radians) = 2πr(adius) • then linear quantity = r(adius) ∙ angular quantity so Δx = r∙Δθ ; v = r∙ω ; a = r∙ also v as f (in rps) ∙ 2π (in radians) /1 rev = ω(in rads/s)

Translational Motion Rotational Motion Dynamics (C 4) Dynamics (8. 4 -6) Forces - F - cause Torques - - cause rotation acceleration when applied at distance, r, to axis containing the CM from axis of rotation =rx. F Inertia – m – tendency to Moment of inertia – I – is resist changes in motion; tendency to resist changes in aka acceleration rotation; aka angular acceler depends on mass in kg & distribution of mass in m 2 N 2 nd. L: ΣF = ma N 2 nd. L for Rot: Σ = I where I = c mr 2 in kgm 2 see pg 208: c depends on object shape & loco axis of rot

Translational Motion Work by Force (C 6) W = F ● Δx & Power = W/Δt = F ● v Rotational Motion Work by Torque (8. 7) W = ● Δθ = (r x F) ● (Δx/r) and P = ● Δθ/Δt = ● ω Also recall: Then comparatively: W = F●Δx is dot product = r x F is a cross product = FΔxcosθ in Nm = r. Fsinθ in m. N so max work when so max torque when F & Δx are parallel r & F are and W = 0 when and = 0 when F & Δx are r & F are parallel makes sense because ll F would cause a, not rotation…

Translational Motion Energy (C 6) KElin = ½mv 2 in kgm 2/s 2 Rotational Motion Energy (8. 7) KErot = ½Iω2 in kgm 2/s 2 Angular Momentum (8. 8) Linear Momentum (C 7) L=Iω p = mv Δp = mΔv or Δmv = ΣFΔt ΔL = IΔω or ΔIω = Σ Δt Law of Conservation of p: Law of Conservation of L: if ΣFext = 0, if Σ ext = 0, then Δp = 0 then ΔL = 0 and pi = pf and Li = Lf

Translational Motion Vectors (C 3) directions can be seen, witnessed, experienced except for a… which is more like rotational… Rotational Motion Vectors (8. 9) directions must be defined n r: out from & to axis of rot n I is scalar – dot prod of r 2 n Δθ, ω, L along axis of rot, by 1 st (easy) RHR n , , ΔL in • same direction as ω; if n ω & turning CCW n ω & turning CW • oppo direction as ω; if n ω & turning CW n ω & turning CCW by 2 nd (harder) RHR

Right Hand Rules for Rotational Motion 1 st RHR: fingers curl in direction of rotation thumb points in direction of Δθ, ω, L 2 nd RHR: fingers point in direction of r palm points in direction of F thumb points in direction of , , ΔL

Translational Motion Rotational Motion Frames of Ref (C 2 -7) Frames of Ref (Appendix C) inertial – not aing, inertial – not rotating, so N’s Laws good non inertial – aing, non inertial – rotating, so feel fictitious force like centrifugal… like Coriolis force • in N. hemi – CCW spin • in S. hemi – CW spin • weak, but present in large air & water masses

Circular Motion (C 5) vs. Rotational Motion (C 8)? ? Circular Motion • revolution about an external axis • caused by a centripetal force • that must be constantly applied • otherwise the object will move off tangent • as Newton’s 1 st Law dictates Ex: stopper on a string Earth’s year about the sun merry-go-round? for the rider…

Circular Motion (C 5) vs. Rotational Motion (C 8)? ? Rotational Motion • rotation about an internal axis; aka spin • caused by a torque • that once applied, • the object will continue with that rotation • as the Law of Conservation of Angular Momentum dictates Ex: Ice skater in “final” spin Earth’s day on its own axis child’s top merry-go-round – for the structure

Types of Velocity (Linear) Velocity – rate at which displacement is covered eq’n: v = Δx/Δt units: m/s Tangential Velocity – rate at which distance is covered as something moves in a circular path – so the distance would amount to some multiple of the circumference of a circle eq’n: v = 2∏r/T, tangent to circle units: m/s Linear & tangential speed measure essentially the same thing, but for an object moving in a circle Angular Velocity (ω) – aka rotational velocity – rate at which something rotates eq’n: ω = Δθ/Δt units: radian/s, rev/s, rpm

Rigid body – any object whose particles maintain the same position relative to each other during motion/rotation. Non rigid examples: For a rigid body, points closer to the axis of rotation • have less tangential speed • same rotational speed Sometimes groups of people try to act like a rigid body: Ex: planes in formation marching Band in parade formation Ice Capades or a Rockettes performance

Rotational Inertia Recall moment of inertia (I = cmr 2) n It is often referred to as rotational inertia – tendency for an object to resist a change in its state of rotation dependent on mass (m) & on distribution of mass (c r 2) n close to axis much less I n far from axis much more I N 2 nd L for Rotation ( = I ) an external net torque ( ) is required to change an object’s rotation - to give it rotational acceleration ( ) n

ex: meter sticks with movable masses pole with movable mass choking up on a bat, club or drumstick straight vs well bent legs

more ex: tightrope walker, with vs without pole long/heavy tail on an animal dinosaurs, kangaroos monkeys, cats various shape objects racing down an incline

When an object slides down a frictionless incline PEtop = KElin at bottom But when an object rolls, it takes some of its NRG just to spin (KErot), so then there’s not as much left to move it down the plane (KElin) PEtop = KEroll at bottom = KErot at bot + KElin at bot But how much of each KErot & KElin does it have? It depends on object’s rotational inertia (I) which depends on object’s distrib of mass: If mass is distributed far from the axis, then more I, so more KErot leaving less KElin to move along ramp, so it loses the race.

But does mass or size (radius) matter? Recall: KErot = ½Iω2 and I = c mr 2 while ω = v/r so KErot = ½ c mr 2 (v/r)2 So radius cancels (doesn’t matter), leaving us with KErot = ½ c mv 2 And when you put it all together for a given object at the top of a ramp: PEtop = KErot at bot + KElin at bot mghtop = ½ c mv 2 + ½mv 2 so mass cancels too! So for a particular h, what determines which object will have the greater v (win), is its smaller “c” value which is only based on shape & axis of rot (p. 208)

Angular Momentum A cool result of an object having angular momentum: the object becomes incredibly stable and balances itself! This is due to the nature of the vectors in rotational motion. They actually act as a force to balance out gravity, which would otherwise topple the object.

Angular Momentum Ex: child’s toy top gyroscope basketball plates on top of a tall rod bicycle wheels

Conservation of Angular Momentum Law of Conservation of Angular Momentum – for an isolated* system the amt of L is a constant * forces to the CM ok since they don’t cause rotation, but no torques allowed! Recall: L = I ω; so for an isolated rotating object, if it redistributes its mass (changes I), then its rotational speed (ω) will change inversely Ex: Rotating Platform with arms out, that’s a larger I, so your ω gets smaller to compensate Other ex: flips in gymnastics & diving spins in figure skating falling cat lands on its feet quarterback twisting during a throw