Rotational Energy

discussion summary practice problems Rotational Energy Discussion Rotational kinetic energy For a system of point bodies K =  1  ∑ mivi2 2 K =  1  ∑ ri2mi  vi2 2 ri2 K =  1 Iω2 2 For an extended body K =  1 ⌠ ⌡ v2 dm 2 K =  1 ⌠ ⌡ v2 r2 dm 2 r2 K =  1 ω2I 2 K =  1 Iω2 2 Moment of inertia I =  ...

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Angular Momentum

discussion summary practice problems Angular Momentum Discussion the idea text L = r × p L = r × mv L = mr × (ω × r) L = mω(r · r) − mr(r · ω) L = mr2ω − 0 L = mr2ω L = Iω Moment of inertia again I = ∑ r2m =  ⌠ ⌡ r2 dm = ρ ⌠ ⌡ r2 dV Conservation of angular momentum ∑ τ = d  L = 0 dt Thus L = constant Translational and rotational quantities compared concept translation connection rotation momentum p = mv L = r × p = mr × v L = ...

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Rotational Equilibrium

discussion summary practice problems Rotational Equilibrium Equilibrium of Extended Bodies Discuss. ∑ τ⟲ = ∑ τ⟳ counter clockwise is positive clockwise is negative Translational and rotational quantities compared concept translation connection rotation equilibrium ∑F = 0 ⇒  ∑ F+x = ∑ F−x ∑ F+y = ∑ F−y ∑ F+z = ∑ F−z ∑ τ = 0 ⇒  ∑ τ+x = ∑ τ−x ∑ τ+y = ∑ τ−y ∑ τ+z = ∑ τ−z Center of Mass The center of mass is computed from the mass distribution. Discrete. rcm =  ∑ miri  = (x̅, y̅, z̅) ∑ mi Continuous. rcm =  1 ⌠⌠⌠ ⌡⌡⌡ r dm = (x̅, y̅, z̅) m Continuous ...

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Rotational Dynamics

discussion summary practice problems Rotational Dynamics Taste and compare… Translational and rotational laws of motion translational rotational 1st An object at rest tends to remain at rest and an object in motion tends to continue moving with constant velocity unless compelled by a net external force to act otherwise. An object at rest tends to remain at rest and an object in rotation tends to continue rotating with constant angular velocity unless compelled by ...

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Rotational Inertia

discussion summary practice problems Rotational Inertia introduction & theory Logic behind the moment of inertia: Why do we need this? Definition for point bodies I = mr2 It’s a scalar quantity (like its translational cousin, mass), but has unusual looking units. [kg m2] Say it, kilogram meter squared and don’t say it some other way by accident. For a collection of objects, just add the ...

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Rotational Kinematics

discussion summary practice problems Rotational Kinematics let’s rap discussion Translational and rotational quantities compared concept translation connection rotation base quantities s, r   s =  θ × r θ   coordinate systems r =  x î + y ĵ x =  y =  r2 =  θ =  r cos θ r sin θ x2 + y2 tan−1 (y/x) r =  r r̂ + θ θ̂ velocity v =  dr/dt v =  ω × r ω =  dθ/dt acceleration a =  dv/dt = d2r/dt2 a =  α × r − ω2r α =  dω/dt = d2θ/dt2 equations of motion v =  x =  v2 =  v0 + at x0 + v0t + ½at2 v02 + 2a(x − x0) ...

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Momentum & Energy

discussion summary practice problems Momentum & Energy Discussion general info two objects (1 and 2), velocities before and after (unprime and prime) conservation of momentum m1v1 + m2v2 = m1v′1 + m2v′2 “conservation of kinetic energy” — not a law, just a statement of a possibility ½m1v12 + ½m2v22 = ½m1v′12 + ½m2v′22 Solve for the velocities after collision. (This is a painful process.) There are two pairs of solutions. v′1 =  (m1 − m2)v1  + 2m2v2 ...

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Conservation Of Momentum

discussion summary practice problems Conservation Of Momentum Discussion start Start with Newton’s third law of motion, toss in the impulse-momentum theorem, and see what happens. +F1 = −F2 +F1Δt = −F2Δt +m1Δv1 = −m2Δv2 +Δp1 = −Δp2 +(p1′ − p1) = −(p2′ − p2) p1′ + p2′ = p1 + p2 ∑p′ = ∑p Or flip it around. ∑p = ∑p′ The total momentum before an interaction is the same as the total momentum after the interaction. ...

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Impulse & Momentum

discussion summary practice problems Impulse & Momentum Discussion what is impulse ? Momentum keeps me going. From the Principia. Quantitas motus est mensura ejusdem orta ex velocitate et quantite materiæ conjunctim.   The quantity of motion is the measure of the same, arising from the velocity and the quantity of matter conjunctly. We now call this quantity momentum. Momentum is resistance to stopping. ...

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