角向运动和直线运动的转化

Suppose that an object is moving in a circle (not necessarily uniform motion) with radius $R$.

We would like to know how to convert between values that describe linear motion (such as velocity, acceleration, and momentum) and values that describe angular motion (such as angular velocity, angular acceleration, and angular momentum).

To start off, let $\theta$ be the angular displacement, and let $x$ be the linear distance displacement.

By the definition of angle measure in radians, we have: $$x=\theta R$$ Now, let us differentiate to find the velocity. Note that $R$ is constant: $$\frac{d}{dt}(x)=R\frac{d}{dt}(\theta)$$ But wait! Clearly, the instantaneous rates of change of the linear and angular displacement are simply the linear and angular velocity, denoted $v$ and $\omega$, respectively: $$v=\omega R$$ Let us differentiate once more. Again, $R$ is a constant: $$\frac{d}{dt}(v)=R\frac{d}{dt}(\omega)$$ Since the derivative of the velocity is the acceleration (denoted $a$), and since the derivative of the angular velocity is the angular acceleration (denoted $\alpha$), we have the following: $$a=\alpha R$$ Really, the core principle to keep in mind is how angle measure in radians is defined (arc length divided by radius). If you know this, you can basically re-derive all of the following results at will.

Also, when working with trickier things, such as rotational kinetic energy, it is helpful to remember that in a sufficiently small time interval, everything is basically moving in linear motion. Thus, one may divide the object into lots of little pieces, add up all of their linear kinetic energies, and then convert back into angular terms with the equations above.