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.

 

[astrokreng]

The reason for the similarities between rotational and linear dynamics lies in the fundamental principles of physics. Both rotational and linear motion are governed by Newton's laws of motion, which state that an object will remain at rest or continue moving with a constant velocity unless acted upon by a force. In rotational motion, this force is known as torque, while in linear motion it is known as force.

Additionally, the concept of inertia, which is the tendency of an object to resist changes in its motion, applies to both rotational and linear motion. In rotational motion, this is known as moment of inertia, while in linear motion it is simply mass.

The mathematical equations that describe rotational and linear motion are also similar because they are derived from the same fundamental principles. In both cases, the equations involve the relationship between force, mass (or moment of inertia), and acceleration.

Furthermore, there is a deep analogy between rotating and moving in a straight line. Both involve the concept of angular velocity, which is the rate at which an object is rotating or moving in a circular path. This is similar to linear velocity, which is the rate at which an object is moving in a straight line. Both angular and linear velocity also have the same units (radians/second or meters/second), further highlighting the connection between the two.

In summary, the similarities between rotational and linear dynamics can be attributed to the fundamental principles of physics and the deep analogy between rotating and moving in a straight line.