Basic Rotational Quantities

In addition to any tangential acceleration, there is always the centripetal acceleration: 除了切向加速度之外，始终存在向心加速度：

The angular displacement is defined by: 角位移定义为： For a circular path it follows that the angular velocity is 对于圆形路径来说，角速度是 and the angular acceleration is 并且角加速度是 where the acceleration here is the tangential acceleration. 其中的加速度是指切向加速度。

Angular Velocity

Angular velocity can be considered to be a vector quantity, with direction along the axis of rotation in the right-hand rule sense. 角速度可以被视为一个矢量，其方向沿旋转轴，符合右手法则的方向。 Vector angular velocity 矢量角速度

For an object rotating about an axis, every point on the object has the same angular velocity. The tangential velocity of any point is proportional to its distance from the axis of rotation. Angular velocity has the units rad/s. 对于绕轴旋转的物体，其上每个点都有相同的角速度。任何一点的切向速度与其到旋转轴的距离成正比。角速度的单位是rad/s。 Angular velocity is the rate of change of angular displacement and can be described by the relationship 角速度是角位移的变化率，可以由关系式描述 and if v is constant, the angle can be calculated from 并且如果v是常量，角度可以从

Description of Rotation

Rotation is described in terms of angular displacement, time, angular velocity, and angular acceleration. Angular velocity is the rate of change of angular displacement and angular acceleration is the rate of change of angular velocity. The averages of velocity and acceleration are defined by the relationships:

Average angular velocity: 平均角速度：

Average angular acceleration: 平均角加速度：

where the Greek letter delta indicates the change in the quantity following it.

A bar above any quantity indicates the average value of that quantity. If α is constant, equations 1, 2, and 3 represent a complete description of the rotation. Equation 4 is obtained by a combination of the others. 任何量上方的横线表示该量的平均值。如果α是常数，方程1、2和3对旋转的描述是完整的。方程4是通过其他方程的组合得到的。

You might want to try a numerical exploration of these equations and see them stated in words.

Rotation Equations

These rotation equations apply only in the case of constant angular acceleration. It is assumed that the angle is zero at t=0 and that the motion is being examined at time t.

*These quantities are assumed to be given unless they are specifically clicked on for calculation.

You can probably do all this calculation more quickly with your calculator, but you might find it amusing to click around and see the relationships between the rotational quantities. It's like a game to see if you can set up a consistent set of parameters. Associated with each active text is a Javascript calculation routine.

Comments on Rotation Calculation

In the example calculation in the section Rotation Equations above, some assumptions were made about the calculation order. The rotation equations represent a complete set of equations for constant angular acceleration rotations, but in certain types of problems, intermediate results must be calculated before proceeding to the final calculation. In the example calculation, you may have to do intermediate calculations, e.g., to establish the final angular velocity, in order to set up the problem you wish to solve, just as if you were working the problem with calculator and paper. In the example calculation, the time, initial angular velocity, and angular displacement were considered given (primary) unless they were being calculated (e.g., in calculating α). If θ is being calculated, then ω is assumed given, so it must be calculated first if you want to specify α. If initial angular velocity is being calculated, then ω is presumed given. If time is being calculated then α is presumed given. These presumptions are included in the Javascript routines.