Rotational Dynamics

Discussion

Taste and compare …

  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 a net external torque to act otherwise.
2nd Acceleration is directly proportional to net force and inversely proportional to mass or
F = m a
  Rotational acceleration is directly proportional to net torque and inversely proportional to moment of inertia or
τ = I α
3rd For 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 laws of motion
concept translation connection rotation
cause of acceleration ∑ F   τ =  r × F ∑ τ  
resistance to acceleration m   I =  ∑ ri2mi = ∫ r2 dm I  
newton's second law ∑ F =  m a     ∑ τ =  I α
Translational and rotational quantities compared

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 =  m a
r × F =  m r × a = m r × (α × r)
r × F =   m α(r · r) − r(r · α) = m α r2 − 0 = mr2 α
τ =  I α

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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