# Kinetic energy of a yoyo

Consider the yoyo above:

The yoyo is constructed from two heavy disks of radius R connected by a light axle of radius r , as shown in the figure below. The total mass of the yoyo is M and its moment of inertia for rotations around the axle can be taken to be $$\frac12MR^2$$

as it is spinning and rotating and travelling downwards i am aware that the total kinetic energy is as follows:

$$\frac12 Iw^2 +\frac12mv^2$$

is the v term wR or wr?

• [...] and its moment of inertia for rotations around the axle can be taken to be [...] It rotates about the centre of the yoyo. – Gert Apr 17 at 16:43
• Find the instantaneous axis of rotation – Aditya Garg Apr 17 at 16:59

$$v$$ would be equal to $$\omega$$ multiplied by the radius of the axle.
In reality the radius of the axle would likely be somewhere in between $$r$$ and $$R$$. The reason being that as the yoyo string wraps around the axle, it would essentially increase the circumference and radius of the axle, as the string wrapped around the axle can be treated as part of the axle in this example.
However, if you were to say that the string around the axle is negligible (or if you are calculating this right at the instant when the yoyo has completely unwound all of the string), then it would be simply $$\omega r$$.