Transcript File

An Overview of Rotational Motion (C8) via a
Comparison to Translational Motion (C2-4,6-7)
Note these essentially interchangeable terms:
linear & translational
angular & rotational
Translational Motion
Rotational Motion
Kinematics (C2)
Kinematics (8.1-3)
 Basic variables
 Basic variables
Δx: displacement (m)
Δθ: angular disp (radians)
v: velocity (m/s)
ω: angular velocity (rad/s)
: angular accel (rad/s2)
a: acceleration (m/s2)
 Constant  eq’ns:
 Constant a eq’ns:
ωf = ωi + Δt
vf = vi + aΔt
Δθ = ωi Δt + ½Δt2
Δx = vi Δt + ½ a Δt2
Δθ = ωf Δt - ½Δt2
Δx = vf Δt - ½ a Δt2
ωf2 = ωi2 + 2Δθ
vf2 = vi2 + 2a 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 (C4)
Dynamics (8.4-6)
Forces - F - cause
Torques -  - cause rotation
acceleration when applied when applied at distance, r,
to axis containing the CM from axis of rotation
=rxF
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
depends on mass in kg &
distribution of mass in m2
N2ndL: ΣF = ma
N2ndL for Rot: Σ = I 
where I = c mr2 in kgm2
see pg 208: c depends on
object shape & loco axis of rot
Translational Motion
Work by Force (C6)
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
= rFsinθ in mN
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 (C6)
KElin = ½mv2 in kgm2/s2
Rotational Motion
Energy (8.7)
KErot = ½Iω2 in kgm2/s2
Angular Momentum (8.8)
Linear Momentum (C7)
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 (C3)
directions can be
seen, witnessed,
experienced
except for a…
which is more like
rotational…
Rotational Motion
Vectors (8.9)
directions must be defined
 r: out from &  to axis of rot
 I is scalar – dot prod of r2


Δθ, ω, L along axis of rot, by
1st (easy) RHR
, , ΔL in
• same direction as ω; if
 ω  & turning CCW
 ω  & turning CW
• oppo direction as ω; if
 ω  & turning CW
 ω  & turning CCW
by 2nd (harder) RHR
Right Hand Rules for Rotational Motion
1st RHR: fingers curl in direction of rotation
thumb points in direction of Δθ, ω, L
2nd 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 rotating,
inertial – not aing,
so N’s Laws good
so N’s Laws good
non inertial – rotating,
non inertial – aing,
so feel fictitious force so feel fictitious force
like Coriolis force
like centrifugal…
• in N. hemi – CCW spin
• in S. hemi – CW spin
• weak, but present in
large air & water masses
Circular Motion (C5) vs. Rotational Motion (C8)??
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 1st Law dictates
Ex: stopper on a string
Earth’s year about the sun
merry-go-round?
for the rider…
Circular Motion (C5) vs. Rotational Motion (C8)??
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 = cmr2)
 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 r2)
 close to axis  much less I
 far from axis  much more I
N 2nd L for Rotation ( = I ) an external net
torque () is required to change an object’s
rotation - to give it rotational acceleration ()

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 mr2
while
ω = v/r
so KErot = ½ c mr2 (v/r)2
So radius cancels (doesn’t matter), leaving us with
KErot = ½ c mv2
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 mv2 + ½mv2
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