Rotational Motion

Coriolis Force

discussion summary practice problems Coriolis Force Discussion Talk, Talk, Talk Gustave Coriolis (1792–1843) France Mémoire sur les équations du mouvement relatif des systèmes de corps. Gaspard-Gustave Coriolis. Journal de l’École polytechnique. Vol. 15 No. 24 (1835) 142–154. Should this be changed to a discussion of radial vs. tangential acceleration? Forces in a rotating coordinate system direction real fictitious radial  ar =  radial ⎛ ⎝ d2r ⎞ ...

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Rolling

discussion summary practice problems Rolling Discussion Rolling without slipping is a combination of translation and rotation where the point of contact is instantaneously at rest. When an object experiences pure translational motion, all of its points move with the same velocity as the center of mass; that is in the same direction and with the same speed v(r) =  vcenter of mass ...

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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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