The person who had my room before me left a few of their Command Hooks up around the room, including a few on the wardrobe and which looks something like this:
One day, I hung a jacket on a hanger on hook 2 and it moved really fast and nearly flew off of the hook. When I put the jacket on hook 1, it didn’t move nearly as fast. Naturally, I wondered what the difference in linear velocity was between the two hooks, and if the difference was really that big.
If we use the hinges as the axis of rotation, hook 1 is 0.10 m from the axis of rotation and hook 2 is 0.38 m from the axis of rotation. It took me approximately 0.60 s to move the door 90° (1.57 radians).
ω = △θ/△t=1.57 rad/0.60s=2.62 rad/s.
Since angular velocity is not dependent on radius, it is the same for both hooks.
The linear velocity is dependent on radius, so it is different for each hook.
v1=r= (2.62 rad/s)(0.10m)= 0.262 m/s
v2= r= (2.62 rad/s)(0.38m) = 0.996 m/s
While that may seem like a relatively small difference in linear velocity, it is apparently a considerable difference for a hanger on a closet door, and is enough to make a difference in whether or not the jacket stays on the door.