Relationship between Angular and Tangential Velocities in Rotating Gears

[AussieDave]

Hello. I have what I hope to be a simple question. It's something that I've come across in several problems in my engineering and physics subjects and I'm not sure exactly what relationship applies. My textbook doesn't seem to be very helpful in the matter.

When there are two circles, I suppose you could call them gears, of different sizes with a belt wrapped around them, what is the relationship between their velocity? I'm talking about both their angular and tangential velocities.

I know that in a rotating disc the angular velocity will be the same at each point on the disk but the tangetial velocity will change as you change your radius from the centre of the disk.

I have a feeling that either the tangential or angular velocity will be the same for these two disks attached by the belt with 1 of them being turned by a motor and, in turn, turning the other.

This isn't a homework problem but I've attached a simple example diagram with simple numbers to illustrate my question. Here it is the smaller disk being turned by the motor at a given angular velocity and I'd like to know what I can therefore infer about the larger disk.

Thank you kindly in advance for your help,

David.

 

[Cyrus]

Find a relationship between the arc lengths.

 

[Shooting Star]

I have a feeling that either the tangential or angular velocity will be the same for these two disks attached by the belt with 1 of them being turned by a motor and, in turn, turning the other.

You are psychic!

Think of the linear speed of the belt, if there is no slipping. After all, that's the only thing which connects the two gears.

 

[tiny-tim]

I know that in a rotating disc the angular velocity will be the same at each point on the disk but the tangetial velocity will change as you change your radius from the centre of the disk.

I have a feeling that either the tangential or angular velocity will be the same for these two disks attached by the belt with 1 of them being turned by a motor and, in turn, turning the other.

Hi David!

A straight answer:

Yes, it's the tangential velocity.

This is because the speed of the belt is the same all the way along the belt. So its speed on one wheel is the same as its speed on the other wheel.

Which is just the tangential velocity!

(btw, it's the same with gears, ie the wheels touching and with no belt - the tangential velocity is the same - except of course that the wheels rotate in opposite directions instead of the same direction.)

 

[AussieDave]

Thank you everyone for your response. My thought was tangential velocity for the very reason you just posted. I was just struggling to find a straight answer in several textbooks. My engineering one has lots of great, tricky questions with nice diagrams but it barely provides any actual theory behind them. The answers also aren't very helpful but I'm getting the instructor's manual.

I was then going to ask the question about the gears but you've solved that too so thank you very much!