mechanics

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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Simple Machines Problems

discussion summary practice problems Problems practice Write something else. A bucket grain elevator is powered by a 150 kW electric motor. Determine its efficiency if it can lift grain to a height of 60 m at a rate of 900 m3/h. Assume the elevator is lifting wheat with a density of 770 kg/m3. Write something different. Write something completely different. numerical problem

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Simple Machines Practice

discussion summary practice problems Practice practice problem 1 Write something else. solution Answer it. practice problem 2 A bucket grain elevator is powered by a 150 kW electric motor. Determine its efficiency if it can lift grain to a height of 60 m at a rate of 900 m3/h. Assume the elevator is lifting wheat with a density of 770 kg/m3. solution Start with ...

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Simple Machines

discussion summary practice problems Discussion In the most general sense, a machine is any device that can be used to perform a task. In the mechanical sense, a machine is a device for transmitting work from one location to another. A bicycle is a machine. The rider does work on the pedals, which in turn do work on the front crank, which ...

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