Rotational Kinetic Energy

The kinetic energy of a rotating object is analogous to linear kinetic energy and can be expressed in terms of the moment of inertia and angular velocity. The total kinetic energy of an extended object can be expressed as the sum of the translational kinetic energy of the center of mass and the rotational kinetic energy about the center of mass. For a given fixed axis of rotation, the rotational kinetic energy can be expressed in the form

旋转物体的动能与线性动能类似，可以表示为转动惯量和角速度的函数。扩展物体的总动能可以表示为质心的平动动能与绕质心的转动动能之和。对于给定的固定旋转轴，转动动能可以表示为

The expressions for rotational and linear kinetic energy can be developed in a parallel manner from the work-energy principle. Consider the following parallel between a constant torque exerted on a flywheel with moment of inertia I and a constant force exerted on a mass m, both starting from rest.

旋转动能和线性动能的表达式可以从功-能原理中平行地推导出来。考虑以下关于作用在飞轮上的恒定力矩（转动惯量为I）与作用在质量为m的物体上的恒定力之间的平行关系，两者均从静止开始。

For the linear case, starting from rest, the acceleration from Newton's second law is equal to the final velocity divided by the time and the average velocity is half the final velocity, showing that the work done on the block gives it a kinetic energy equal to the work done. For the rotational case, also starting from rest, the rotational work is τθ and the angular acceleration α given to the flywheel is obtained from Newton's second law for rotation. The angular acceleration is equal to the final angular velocity divided by the time and the average angular velocity is equal to half the final angular velocity. It follows that the rotational kinetic energy given to the flywheel is equal to the work done by the torque.

对于直线情况，从静止开始，根据牛顿第二定律，加速度等于末速度除以时间，平均速度是末速度的一半，这表明作用在块上的功使其获得的动能等于所做的功。对于旋转情况，同样从静止开始，旋转功为τθ，施加给飞轮的角加速度α由旋转的牛顿第二定律得到。角加速度等于末角速度除以时间，平均角速度等于末角速度的一半。由此可知，施加给飞轮的旋转动能等于扭矩所做的功。