advanced rotational dynamics for ap physics

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


+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)


+Examples of Rotation+Review of Introductory Rotational Dynamics





+Examples of Rotation

+Review of Introductory Rotational Dynamics




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



+Combined translation and rotation



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



+Combined translation and rotation



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



+Combined translation and rotation+Rolling without slipping (pure rolling)



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




+Rolling with slipping

+Rotational form of Conservation Laws


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)





+Collisions involving objects having Moment of

Inertia





+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!

advanced rotational dynamics for ap physics

Documents

direction of rotation

large rotational velocitym31

moment of inertiaexamples

fixed axisgear

direction of omegai

direction of earths

torque ccw

torque net