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Advanced Rotational Dynamics for AP Physics

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## Advanced Rotational Dynamics for AP Physics

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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**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**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**steam driven power plant turbine:imagine this thing rotating**at 60 hzgenerating your electricity**electric motors are backwards connected generators:they are**still mechanical rotators 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 αΔθ • 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**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**• 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)**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