Rotational & linear dynamics: why so similar?

[Christopher M]

I'm not an advanced physics student -- I've only taken the basic year -- but I'm curious about a conceptual issue and wonder if someone could give me a satisfying explanation. There are obviously pretty tight parallels between basic linear kinematics/dynamics and the rotational equivalents -- so (F = ma) is analogous to (torque = moment of inertia * rotational acceleration); and the basic equations of rotational kinematics similarly have exact analogs in the linear kinematic equations.

So why is this? It doesn't actually seem obvious that there should be a direct rotational analog to "momentum" that functions in exactly the same way, mathematically, that momentum does in linear motion. Can anyone help me out with an explanation -- is there some deeper analogy between rotating and moving in a straight line?

Again, I realize this isn't the most sophisticated question, so thanks in advance for helping satisfy my untutored curiosity.

 

[D H]

You'll find the reason for the analogy next year when you take classical mechanics. If you are truly interested, take a look at the text for that class.

You will also find that the analogy isn't perfect. Mass is a scalar while inertia is a 2nd order tensor. Moreover, the inertia tensor is constant in a frame rotating with the object, but not from the perspective of an inertial frame. This means that rotational equations of motion are most easily expressed from the perspective of a rotating frame. I assume you are aware that one must conjure up fictitious forces (centrifugal and coriolis forces) to make Newton's laws appear to be valid in a rotating frame. The same applies to rotational motion. The equation τ = Iω is, in general, incorrect. It ignores the effects of doing physics in a rotating frame. To date you have only been shown examples where the tensorial and rotating frame nature of the equations of motion can be ignored.