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Advanced Rotational Dynamicsfor AP Physics
+Common Examples of Rotation+Review of Introductory Rotational Dynamics
+Combined translation and rotation+Rolling without slipping
+Rolling with slipping+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia
Advanced Rotational Dynamicsfor AP Physics
+Common Examples of Rotation+Review of Introductory Rotational Dynamics
+Combined translation and rotation+Rolling without slipping
+Rolling with slipping+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia
Advanced Rotational Dynamicsfor AP Physics
+Common Examples of Rotation+Review of Introductory Rotational Dynamics
+Combined translation and rotation+Rolling without slipping
+Rolling with slipping+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia
Advanced Rotational Dynamicsfor AP Physics
+Common Examples of Rotation
+Review of Introductory Rotational Dynamics+Combined translation and rotation
+Rolling without slipping+Rolling with slipping
+Rotational form of Conservation Laws +Collisions involving objects having Moment of Inertia
Examples of rotation: Earth
To find the direction of omega, apply the right hand rule as follows:
With fingers curling in direction of rotation, the thumb gives direction of omega
i.e. direction of earth’s omega is upward
rotation of the sky
star trails
centered on Polaris
rotate once a day
rotation can be fun
sometimes the goal is rotational equilibrium
• 1st condition for equilibrium:
Fnet = 0
• 2nd condition for equilbrium:
torque net = 0
• i.e., torque ccw = torque cw
rotational equilibrium again
sometimes the goal is large rotational velocity
M31 rotates once every few hundred million years
Katrina rotated
a generic kidney shaped object rotates about a fixed axis:
a thing of beauty is a joy forever!
a turbine (driven by moving fluid) rotates about a fixed axis
a jet engine rotates about a fixed axis
steam driven power plant turbine:imagine this thing rotating at 60 hz
generating your electricity
electric motors are backwards connected generators:
they are still mechanical rotators about a fixed axis
electric motors power our elderly friends’ scooters
and nuclear subs also(when they are submerged)
gears rotate about a fixed axis
gear attached to an electric motor(sounds like a good idea to me)
nanotechology electric motor gear (the gear teeth are smaller than a red blood cell)
rotating about a fixed axis, imaged with an electron microscope
(end of examples of rotation)
Advanced Rotational Dynamicsfor AP Physics
+Examples of Rotation+Review of Introductory Rotational Dynamics
+Combined translation and rotation+Rolling without slipping
+Rolling with slipping+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia
Advanced Rotational Dynamicsfor AP Physics
+Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation+Rolling without slipping
+Rolling with slipping+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia
Advanced Rotational Dynamicsfor AP Physics
+Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation+Rolling without slipping
+Rolling with slipping+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia
Rotational Kinematics
• Θ = angular position wrt arbitrary origin
• Δθ = angular displacement (rad)
• ω = Δθ / Δt = dθ / dt (rad/s)
• α = Δ ω / Δt = d ω / dt = d2θ / dt2 (rad/s2)
• s = r θ• v = r ω• a = r α
α = constant implies
• Δθ = ω0t + ½ αt2
• ω = ω0 + α t
• ω2 = ω02 + 2 α Δθ
• Δθ = ½ (ω0 + ω) t
• If α is variable, you need calculus
Intro Rotational Dynamics
• Г = τ = r x F
• I = Σ mi ri2 (collection of point masses)
= ∫ r2 dm (continuous matter distribution)
• I total = I 1 + I 2 + I 3 + … (composite object)
• Fnet = ma becomes Гnet = τnet = I α
Advanced Rotational Dynamicsfor AP Physics
+Common Examples of Rotation+Review of Introductory Rotational Dynamics
+Combined translation and rotation+Rolling without slipping
+Rolling with slipping+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia
Advanced Rotational Dynamicsfor AP Physics
+Common Examples of Rotation+Review of Introductory Rotational Dynamics
+Combined translation and rotation+Rolling without slipping
+Rolling with slipping+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia
Advanced Rotational Dynamicsfor AP Physics
+Common Examples of Rotation+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping+Rolling with slipping
+Rotational form of Conservation Laws +Collisions involving objects having Moment of Inertia
Earth revolves (translation of cm) and rotates about cm
Look Ma, no hands
combined translation and rotation
Ktotal = Ktranslational + Krotational
= Kof the cm + Karound the cm
= ½ mv2 + ½ I ω2
Advanced Rotational Dynamicsfor AP Physics
+Common Examples of Rotation+Review of Introductory Rotational Dynamics
+Combined translation and rotation+Rolling without slipping
+Rolling with slipping+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia
Advanced Rotational Dynamicsfor AP Physics
+Common Examples of Rotation+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping+Rolling with slipping
+Rotational form of Conservation Laws +Collisions involving objects having Moment of Inertia
Advanced Rotational Dynamicsfor AP Physics
+Common Examples of Rotation+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping+Rolling with slipping
+Rotational form of Conservation Laws +Collisions involving objects having Moment of Inertia
linear and angular velocity and acceleration are proportional
rolling without slipping in our golden years
rolling without slipping
• v = ω r (use with energy conservation)
• atangential = α r (use with 2nd laws)
• friction acts, but does no work
• energy conserved as Wnc = 0
Advanced Rotational Dynamicsfor AP Physics
+Common Examples of Rotation+Review of Introductory Rotational Dynamics
+Combined translation and rotation+Rolling without slipping (pure rolling)
+Rolling with slipping+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia
Advanced Rotational Dynamicsfor AP Physics
+Common Examples of Rotation+Review of Introductory Rotational Dynamics
+Combined translation and rotation+Rolling without slipping (pure rolling)
+Rolling with slipping+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia
Advanced Rotational Dynamicsfor AP Physics
+Common Examples of Rotation+Review of Introductory Rotational Dynamics
+Combined translation and rotation+Rolling without slipping (pure rolling)
+Rolling with slipping+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia
linear and angular velocities and accelerations are independent,
i.e., he’s not getting very much bang (v) for his buck (ω)
don’t try this at home
rolling with slipping
• v ≠ ω r• atangential ≠ α r
• apply Fnet = ma to find atangential of the cm• apply Гnet = Iα to find α around the cm
• to compute t where pure rolling sets in, set a(t) = α(t) r, where a(t) and α(t) are solutions of force and torque equations
Advanced Rotational Dynamicsfor AP Physics
+Common Examples of Rotation+Review of Introductory Rotational Dynamics
+Combined translation and rotation+Rolling without slipping
+Rolling with slipping+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia
Advanced Rotational Dynamicsfor AP Physics
+Common Examples of Rotation+Review of Introductory Rotational Dynamics
+Combined translation and rotation+Rolling without slipping
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia
Advanced Rotational Dynamicsfor AP Physics
+Common Examples of Rotation+Review of Introductory Rotational Dynamics
+Combined translation and rotation+Rolling without slipping
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia
definition of angular momentum:
• L = r x p = r x mv (point m moving at v)
• L = I ω (rigid body with moment I)
law of motion in terms of L:
torque net = ΔL/ Δt = dL / dt
Conservation of angular momentum:
if torque net = 0, then dL / dt = 0
and so, L = constant
so, I initial ω initial = I final ω final (ice skater)
easier said than done
The agony of defeat(net torque ≠ 0, L not conserved)
The thrill of victory(net torque = 0, L conserved)
Conservation of energy, including rotation
Simply allow for rotational
as well as translational K
Conservation of linear momentum
…is not affected by rotational motion, but applies to the cm of the body as before
(i.e., if Fnet,ext = 0 on system. That is,
if something is nailed down and rotates, linear momentum is not conserved, but L can be)
Advanced Rotational Dynamicsfor AP Physics
+Common Examples of Rotation+Review of Introductory Rotational Dynamics
+Combined translation and rotation+Rolling without slipping
+Rolling with slipping+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia
Advanced Rotational Dynamicsfor AP Physics
+Common Examples of Rotation+Review of Introductory Rotational Dynamics
+Combined translation and rotation+Rolling without slipping
+Rolling with slipping+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia
Advanced Rotational Dynamicsfor AP Physics
+Common Examples of Rotation+Review of Introductory Rotational Dynamics
+Combined translation and rotation+Rolling without slipping
+Rolling with slipping+Rotational form of Conservation Laws
+Collisions involving objects having Moment of
Inertia
Advanced Rotational Dynamicsfor AP Physics
+Common Examples of Rotation+Review of Introductory Rotational Dynamics
+Combined translation and rotation+Rolling without slipping
+Rolling with slipping+Rotational form of Conservation Laws
+Collisions (and other systems)
involving objects having Moment of
Inertia
…this is rather important on the AP Exam…
Unfortunately, few common practical applications exist for this type of problem.
They are simply artificially contrived
Any or all 3 conservation laws of energy, linear momentum, and angular
momentum may be satisfied!You have to figure out which ones are
operating, if they don’t tell you!
• If Wnc = 0, then Einitial = Efinal
• If Fnet,ext = 0, then pinitial = pfinal
• If Гnet = 0, then Linitial = Lfinal
The End
That’s it! Now go make a 5!