Rotational Dynamics

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Rotational Dynamics

Taste and compare…

Translational and rotational laws of motion
translationalrotational
1stAn 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 a net external torque to act otherwise.
2ndAcceleration is directly proportional to net force and inversely proportional to massor
F = ma
angular acceleration is directly proportional to net torque and inversely proportional to moment of inertia or
τ = Iα
3rdFor every action there is an equal and opposite reaction. (Here action and reaction refer to forces.)For every action there is an equal and opposite reaction. (Here action and reaction refer to torques.)
Translational and rotational quantities compared
concepttranslationconnectionrotation
cause of acceleration∑F τ = r × F∑ τ 
resistance to accelerationm I = ∑ ri2mi = ∫ r2 dmI 
newton’s second law∑F = ma  ∑ τ = Iα

Start with the equation for Newton’s second law. Cross radius into both sides and then play around with the algebra. (This requires some unusual identities that I’ve seen before, but never really learned.)

∑F = ma
∑r × F = mr × a
∑r × F = mr × (α × r)
∑r × F = mα(r · r) − r(r · α)
∑r × F = mαr2 − 0
∑r × F = mr2α
∑τ = Iα

In the end, we get an analogous formula for Newton’s second law of rotation with two new quantities: torque, the rotational equivalent to force

τ = r × F

and moment of inertia, the rotational equivalent to mass

I = ∑ ri2 mi = 
r2 dm

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